While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
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= Concept Map =
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===Concept Map===
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__FORCETOC__
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{{#drawio:mmTriangles|interactive}}
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<mm>[[Triangles .mm|flash]]</mm>
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= Textbook =
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===Learning Objectives===
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To add textbook links, please follow these instructions to:
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* Identifying a triangle and understanding its formation
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([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage])
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* Recognizing parameters related to triangles
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* Understanding different formation of triangles based on sides and angles
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* Establishing relationship between parameters associated with triangles
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=Additional Information=
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===Teaching Outlines===
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==Useful websites==
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# [http://www.mathsisfun.com/triangle.html All about triangles]
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This is a reference website for types and classification off triangles
A good website for quick reference of all theorems in geometry. Suitable for both students and teachers.
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# Click [http://karnatakaeducation.org.in/KOER/en/index.php/Classification_of_triangles here] for notes on types of triangles.
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# Click [http://karnatakaeducation.org.in/KOER/en/index.php/Classification_of_triangles here] for notes on types of triangles.
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==== Concept #: Formation of a triangle, elements of a triangle and its measures ====
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The triangle is the basic geometrical figure that allows us to best study geometrical shapes. A quadrilateral can be partitioned into two triangles, a pentagon into three triangles, a hexagon into four triangles, and so on.These partitions allow us to study the characteristics of these figures. And so it is with Euclidean geometry—the triangle is one of the very basic element on which most other figures depend. Here we will be investigating triangles and related its properties
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==Reference Books==
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===== Activities # =====
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= Teaching Outlines =
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====== [[Formation of a triangle]] ======
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Introducing formation of a shape with least number of lines and the space enclosed by these lines form a geometric shape.The key geometric concepts that are related with this are explained.
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==Concept #1 A triangle and its basic properties==
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====== [[Measures in triangle|Elements and measures in triangle]] ======
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===Learning objectives===
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The components that make a triangle are investigated. Measuring these components gives a better understanding of properties of triangles. Relation between these components are conceptualized.
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# A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments.
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# It is the polygon with the least number of sides.
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# A triangle can be defined as a polygon which has 3 sides, 3 angles and 3 vertices.
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# The sum of any two sides is always greater than the third side.
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# The angle opposite to longest side is the largest.
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# The angles inside the triangle are its interior angles.
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# The sum of all 3 interior angles in any triangle is always 180 degrees which is called the angle sum property of a triangle.
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# The external angle of a triangle is always equal to the sum of its two opposite interior angles.
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===Notes for teachers===
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====== [[Interior and exterior angles in triangle]] ======
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[''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.'']
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Interior angles are angles that are formed with in the closed figure by the adjacent sides. An exterior angle is an angle formed by a side and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles.
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# A triangle PQR consists of all the points on the line segment PQ,QR and RP. The three line segments, PQ, QR and RP that form the triangle PQ, are called the sides of the triangle PQR.
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[[Category:Triangles]]
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# A triangle has three angles. In figure, the three angles are ∠PQR ∠QRP and ∠RPQ
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[[Category:Class 10]]
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# A triangle has six parts, namely, three sides,PQ QRand RP.Three angles ∠PQR ∠QRP and ∠RPQ. These are also known as the elements of a triangle.
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# The point of intersection of the sides of a triangle is known as its vertex. In figure, the three vertices are P, Q and R. In a triangle, an angle is formed at the vertex. Since it has three vertices, so three angles are formed. The word triangle =tri + angle ‘tri’ means three. So, triangle means closed figure of straight lines having three angles.
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===Activity No # 1 Make your triangle ===
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==== Concept #: Types of triangles based on sides and angles ====
Variations in elements that make a triangle results in distinct triangles. Recognizing these variations helps in interpreting changes that are possible with in a triangle.
# What is the measure of the marked exterior angle?
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# What is the sum of two opposite interior angles?
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# What can you infer ?
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* Evaluation :
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# Reason out as to why external angle is equal to sum of opposite interior angles of a triangle.
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* Question Corner:
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# Try to prove external angle theorem using linear pair concept and angle sum property of a triangle
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==Concept # 2 - Types of triangles==
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==== Concept #: Theorems and properties ====
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===Learning objectives===
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Properties of triangles are logically proved by deductive method. Relations ships between angles of a triangle when a triangle is formed are recognized and understood.
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# Triangles are classified into different types depending on their measures of sides and angles.
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# Based on sides they can be scalene if all sides are of different lengths, isosceles if two sides are equal and equilateral if all 3 sides are of same length.
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# Based on angles, the triangles are classified as Acute angled triangle if all of its angles are acute angled, Obtuse angled if any one of its angles is obtuse, right angled if one of its angles is 90 degrees and as equiangular if all its 3 angles are equal to 60 degrees.
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# An equilateral triangle is equiangular as well.
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===Notes for teachers===
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''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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* Estimated Time: 40 minutes.
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* Materials/ Resources needed:Laptop, geogebra file, projector and a pointer.
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* Prerequisites/Instructions, if any:
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# The students should have prior knowledge of plane figures, triangles, vertex, angles and sides of triangles.
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# They should know the types of angles and measuring angles.
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# They should know that triangles are classified depending on their side lengths as well as angle measurements.
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* Multimedia resources:Laptop
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* Website interactives/ links/ Geogebra Applets:
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* Process (How to do the activity):
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# Begin a discussion like : “ All triangles have ....” .
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# Show them different triangles by moving the vertices.
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# Let students discuss what they noticed about these different triangles (e.g. different side lengths).
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# The teacher will lead the discussion with the questions: “What did you notice about the length of the sides?”, “Do they all have the same length?”
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# Through this discussion help the students develop vocabulary and definitions for the following: equilateral triangle, isosceles triangle, and scalene triangle.
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* Developmental Questions (What discussion questions): # What did you notice about the sides of the triangles ?
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# How many different side lengths can a triangle have? (hint: all 3 sides equal or 2 sides equal all different )
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# What type of a triangle is this ? Why ?
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* Evaluation (Questions for assessment of the child):
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# Identify the different types of triangles from the following images:
Interior angles of a triangle are in relation and also determine the type of angles that can forms a triangle. This also helps in determining an unknown angle measurement.
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# If it is an equilateral triangle, then it has ___ sides that are of the same length.
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# If it is an isosceles triangle, then it has ___ sides that are of the same length.
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# If it is a scalene triangle, then it has ___ sides that are of the same length
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# Can a scalene triangle also be a right-angled triangle ? If yes can you draw one ?
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===Activity No # ===
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'''Video resource:''' Classroom activity for angle sum property of Triangle by NCERT
* '''Evaluation (Questions for assessment of the child)'''
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* '''Question Corner'''
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==Concept #3 - Properties and Theorems: ==
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{{Youtube|BRDAXvQlzt0
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===Learning objectives===
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}}
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# If two sides of a triangle are congruent, then the angles opposite the sides are congruent. This is called isosceles triangle theorem.
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# Conversely if two angles of a triangle are congruent then the sides opposite the congruent angles are congruent.
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# An equilateral triangle is also equiangular.
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# The pythogoras theorem which states that the square of the hypotenuse of a right angled triangle is equal to the sum of the squares of the other two sides.
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# Thales theorem states that a line drawn parallel to any side of the triangle divides the other two sides proportionally.
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===Notes for teachers===
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# The GeoGebra file below verifies the Thales theorem
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{{#widget:YouTube|id=Y-6yYsuGLoc}}
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# Discovered by Pythagoras, a Greek mathematician and philosopher who lived between approximately 569 BC and 500 BC. Pythagoras' Theorem states that: In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. That is:
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[[Image:KOER%20Triangles_html_m2f096af4.png]]
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[[Image:KOER%20Triangles_html_4bd439df.png]]
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'''Pythagoras' Theorem in Three Dimensions'''
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====== [[Angle sum property of a Triangle]] ======
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A three-dimensional object can be described by three measurements - length, width and height.
Interior angle and the corresponding angle form a linear pair. This exterior angle in relation to the remote interior angles and their dependencies are deducted with the theorem.
# The teacher can reiterate a triangle and its elements.
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# Show them the geogebra file and explain the thales theorm or the basic proportionality theorm which states that a line segment drawn parallel to one side of a triangle divides the other two sides proportionally.
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# She can prove the theorm for different measurements by moving the vertices of the triangle and also by moving D, to change the position of parallel line.
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# Also she can use the same file to explain the converse theorm which states that a line segment that divides any two sides of a triangle proportionally is parallel to the third side.
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# While proving converse theorm, she can show that since the corresponding angles <ADE = <ABC and <AED = <ACB, the line DE that divides AB and AC proportionally is infact parallel to BC.
# What does the basic proportonality theorm state ?
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# Name the parallel line from file ?
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# DE is parallel to which side of the triangle ?
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# What are the lengths of AD and DB ?
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# What are the measures of AE and EC ?
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# What is the ratio AD/DB ? and that of AE/EC ?
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# Are the two proportioanal ?
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# What are the areas of triangles ADE and ABC ?
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# What is their ratio ?
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# Is AD/DB = AE/EC = (area ADE)/(area ABC).
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# Recall thales theorm.
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'''Part B:'''
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==== Concept #: Construction of triangles ====
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# What does the converse of thales theorm state ?
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Constructing geometric shapes to precision using a scale and a compass helps in understanding of properties of the shape. Constructing geometric shapes with minimum number of parameters enhances thinking skills.
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# Name the line that divides the two sides AB and AC sides of the triangle proportionally.
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# How can we prove that DE is parallel to BC ?
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# Recall transversal and parallel lines concepts /
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# Identify the transversal ?
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# Identify the corresponding angles.
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# What are the measures of corresponding angles ?
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# Are <ADE and <ABc equal ?
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# Also are <AED = <ACB ?
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# What does equality of correspondng angles indicate ?
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# What is your inference ?
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* Evaluation (Questions for assessment of the child):
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# Recall the basic proportionality theorm and its converse.
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* Question Corner:
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# Formulate any three of your own questions which can be solved using the above theorems.
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===Activity No # 2. Pythogoras theorem===
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The following constructions are based on three essential parameters that are required for construction following the SSS, SAS and ASA theorems
* Evaluation (Questions for assessment of the child):
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* Question Corner:
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= Hints for difficult problems =
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===== Activities # =====
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= Project Ideas =
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====== [[Construction of a triangle with 3 sides|Construction of a triangle with three sides]] ======
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Investigating formation of a unique triangle with the given parameters as the three sides. Constriction follows SSS congruence rule.
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= Math Fun =
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====== [[Construction of a triangle with two sides and included angle|Construction of a triangle with two sides and an angle]] ======
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Construing of a triangle when two of the sides and an angle of a triangle are known and recognizing the role of the given angle, this construction follows SAS congruence rule for the given parameters.
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'''Usage'''
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====== [[Construction of a triangle with two angles and included side]] ======
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Construction of a triangle when two angles of a triangle and a side are known and understanding the side given can only be constructed between the two angles to form a unique triangle. Construction follows ASA congruence rule.
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Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
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====== [[Construction of a right triangle with a side and hypotenuse|Construction of a right angled triangle]] ======
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Right angle is one of the angles of the triangle the and assimilating other parameters that are required to complete construction. Construction of a triangle based on RH congruence rule.
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====== [[Construction of a triangle with side, angle and sum of other two sides|Construction of a triangle with a side, an angle and sum of two sides]] ======
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The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. Construction of a triangle with given parameters sum of two sides and an angle follows SAS congruence rule.
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====== [[Construction of a triangle with side, angle and difference of other two sides|Construction of a triangle with a side, an angle and difference to two sides]] ======
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Difference of two sides and an angle are parameters with which a triangle construction is possible, this construction of triangle follows SAS congruence rule.
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====== [[Construction of a triangle with perimeter and base angles]] ======
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Constructing a unique triangle with perimeter and with other two parameters the base angles of the triangle follows construction on a triangle using SAS congruence rule.
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==== Concept #: Concurrency in triangles ====
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A set of lines are said to be concurrent if they all intersect at the same point. In the figure below, the three lines are concurrent because they all intersect at a single point P. The point P is called the "point of concurrency". This concept appears in the various centers of a triangle.
Rays and line segments may, or may not be concurrent, even when not parallel.
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In a triangle, the following sets of lines are concurrent:
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The three medians.
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The three altitudes.
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The perpendicular bisectors of each of the three sides of a triangle.
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The three angle bisectors of each angle in the triangle.
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The medians, altitudes, perpendicular bisectors and angle bisectors of a triangle are all concurrent lines. Their point of intersections are called centroid, orthocentre, circumcentre and incentre respectively. Concurrent lines are especially important in triangle geometry, as the three-sided nature of a triangle means there are several special examples of concurrent lines, including the centroid, the circumcenter and the orthocenter.
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These points of concurrencies, orthocenter, centroid, and circumcenter of any triangle are collinear that is they lie on the same straight line called the Euler line.
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======Activities #======
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======[[Exploring concurrent lines from given surroundings]]======
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Interactive activity to introduce concurrent lines using examples from our surroundings.
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=====Concept #: Concurrency of medians in triangles.=====
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Median of a triangle is a line segment from a vertex to the midpoint of the opposite side. A triangle has three medians. Each median divides the triangle into two smaller triangles of equal area. The medians of a triangle are concurrent and the point of concurrence is called the centroid. The centroid is always inside the triangle. The centroid is exactly two-thirds the way along each median. i.e the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.
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From Latin: centrum - "center", and Greek: -oid -"like" The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter. Refer to the figure . Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip. Once you have found the point at which it will balance, that is the centroid. In the diagram, the medians of the triangle are shown as dotted blue lines.
======[[Marking centroid of a triangle|Marking centroid of the triangle]]======
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This is a hands on activity to explore concurrent lines formed in a triangle when vertices are joined to the midpoints of the opposite side.
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======[[Medians and centroid of a triangle]]======
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The centroid of a triangle is where the three medians intersect. This activity will show you how to find the centroid and you’ll explore several geometric relationships related to centroid and medians.
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=====Concept #: Concurrency of altitudes in triangles=====
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The distance between a vertex of a triangle and the opposite side is called the altitude of the triangle. Altitude also refers to the length of the segment. Altitudes can be used to compute the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. A triangle has 3 altitudes. The intersecting point of 3 altitudes of a triangle is known as orthocentre of the triangle. This point may be inside, outside, or on the triangle. If the triangle is obtuse, it will be outside. If the triangle is acute, the orthocentre is inside the triangle. The orthocenter on a right triangle would be directly on the 90° vertex. From Greek: orthos - "straight, true, correct, regular" The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.
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======Activities #======
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======[[Altitudes and orthocenter of a triangle]]======
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An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side. A triangle can have three altitudes. Point of intersection of these lines for different types of triangles is explored.
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=====Concept #: Concurrency of Perpendicular bisectors in a triangle=====
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The perpendicular bisector of a triangle is the perpendicular drawn to a line segment which divides it into two equal parts. The point where the three perpendicular bisectors of a triangle meet is called the circumcentre of a triangle. The circumcentre of a triangle is equidistant from all the three sides vertices of the triangle. This common distance is the crcumradius. The circumcentre is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices. The circumcentre of a triangle lies inside or on a side or outside the triangle according as the triangle is acute or right angled or obtuse. The circumcentre of a right angled triangle is the mid-point of its hypotenuse. Latin: circum - "around" centrum - "center"
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One consequence of the Perpendicular Bisector Theorem is that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
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======Activities #======
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[[Perpendicular bisectors and circumcenter of a triangle|'''Perpendicular bisectors and circumcenter of a triangle''']]
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Circumcentre for different types of triangles is investigated with this activity and this further explores several geometric relationships related to the circumcentre and perpendicular bisectors.
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=====Concept #: Concurrency of angle bisectors in triangles.=====
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The ray which bisects an angle is called the angle bisector of a triangle. The point of concurrence of angle bisectors of a triangle is called as incentre of the triangle. The incentre always lies inside the triangle. The distance from incentre to all the sides are equal and is referred to as inradius. The circle drawn with inradius is called incircle and touches all sides of the triangle.
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If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.
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======Activities #======
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======[[Angular bisectors and incenter of a triangle]]======
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The intersecting point of three lines which are the bisectors of three angles of a triangle that is the incenter and it's properties are examined.
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==== Concept #: [[Similar and congruent triangles]] ====
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Two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection. If the objects also have the same size, they are congruent. Two triangles are said to be congruent to one another only if their corresponding sides and angles are equal to one another
The concept of Thales theorem has been introduced in similar triangles. If the given two triangles are similar to each other then,
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* Corresponding angles of both the triangles are equal
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* Corresponding sides of both the triangles are in proportion to each other
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==== Solved problems/ key questions (earlier was hints for problems) ====
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===Additional Resources===
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==== Resource Title and description ====
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1.[http://www.mathopenref.com/tocs/triangletoc.html Triangles]- This resource contain all information related to triangle like definition, types of triangle, perimeter, congruency etc.,
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2. [http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIA Triangles]- This helps to know the aplication of geometry in our daily life, it contain videos and interactives.
4. [http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.SIM Similarity and Congruence]
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5. The below video gives information about Angle Sum Property by Gireesh K S and Suchetha S S
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{{#widget:YouTube|id=GfP8M5GwcdQ}}
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====OER====
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* Web resources:
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** [https://en.wikipedia.org/wiki/Triangle Wikipedia, the free encyclopedia] : The website gives a comprehensive information on triangles from basics to in depth understanding of the topic.
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** [https://jsuniltutorial.weebly.com/ix-triangles.html cbsemathstudy.blogspot.com] : The website contains worksheets that can be downloaded. Worksheets for other chapters can also be searched for which are listed based on class.
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* Books and journals
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* Textbooks
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** NCERT Textbooks – [http://ncert.nic.in/textbook/textbook.htm?iemh1=7-15 Class 9]
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* Syllabus documents
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====Non-OER====
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* Web resources:
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** [https://www.brighthubeducation.com/middle-school-math-lessons/39674-triangle-properties-and-angles/ Bright hub education] : The website describes a lesson plan for introducing triangles and lists classroom problems at the end of the lesson that can be solved for better understanding.
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** [http://www.cpalms.org/Public/PreviewResourceLesson/Preview/40261 CPALMS] : The website contains lesson plan, activities and worksheets associated with triangles.
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** [https://www.urbanpro.com/cbse-class-9-maths-construction UrbanPro] : This website gives downloadable worksheets with problems on triangle construction.
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** [https://schools.aglasem.com/59755 AglaSem Schools] : This website lists important questions for math constructions.
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** [http://www.nios.ac.in/media/documents/SecMathcour/Eng/Chapter-12.pdf National Institute of Open Schooling]: This website has good reference notes on concurrency of lines in a triangle for both students and teachers.
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** [http://www.mathopenref.com/ Math Open Reference] : This website gives activities that can be tried and manipulated online for topics on geometry.
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* Books and journals
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* Textbooks:
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** Karnataka Govt Text book – Class 8 : [http://ktbs.kar.nic.in/New/website%20textbooks/class8/8th-english-maths-1.pdf Part 1] , [http://ktbs.kar.nic.in/New/website%20textbooks/class8/8th-kannada-maths-2.pdf Part 2]
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* Syllabus documents (CBSE, ICSE, IGCSE etc)
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=== Projects (can include math lab/ science lab/ language lab) ===
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'''''Laboratory Manuals''' - Mathematics : [https://ncert.nic.in/pdf/publication/sciencelaboratorymanuals/classIXtoX/mathematics/lelm402.pdf Click here] to refer activity 15,16,18 and 20'' which explains the properties of Triangle.
Identifying a triangle and understanding its formation
Recognizing parameters related to triangles
Understanding different formation of triangles based on sides and angles
Establishing relationship between parameters associated with triangles
Teaching Outlines
Concept #: Formation of a triangle, elements of a triangle and its measures
The triangle is the basic geometrical figure that allows us to best study geometrical shapes. A quadrilateral can be partitioned into two triangles, a pentagon into three triangles, a hexagon into four triangles, and so on.These partitions allow us to study the characteristics of these figures. And so it is with Euclidean geometry—the triangle is one of the very basic element on which most other figures depend. Here we will be investigating triangles and related its properties
Introducing formation of a shape with least number of lines and the space enclosed by these lines form a geometric shape.The key geometric concepts that are related with this are explained.
The components that make a triangle are investigated. Measuring these components gives a better understanding of properties of triangles. Relation between these components are conceptualized.
Interior angles are angles that are formed with in the closed figure by the adjacent sides. An exterior angle is an angle formed by a side and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles.
Concept #: Types of triangles based on sides and angles
Variations in elements that make a triangle results in distinct triangles. Recognizing these variations helps in interpreting changes that are possible with in a triangle.
Video resource: Explanation of Types of Triangles by NCERT.
A triangle can be drawn with different measures of angles which also determine the kind of triangle formed.
Concept #: Theorems and properties
Properties of triangles are logically proved by deductive method. Relations ships between angles of a triangle when a triangle is formed are recognized and understood.
Interior angles of a triangle are in relation and also determine the type of angles that can forms a triangle. This also helps in determining an unknown angle measurement.
Video resource: Classroom activity for angle sum property of Triangle by NCERT
Interior angle and the corresponding angle form a linear pair. This exterior angle in relation to the remote interior angles and their dependencies are deducted with the theorem.
Constructing geometric shapes to precision using a scale and a compass helps in understanding of properties of the shape. Constructing geometric shapes with minimum number of parameters enhances thinking skills.
The following constructions are based on three essential parameters that are required for construction following the SSS, SAS and ASA theorems
Construing of a triangle when two of the sides and an angle of a triangle are known and recognizing the role of the given angle, this construction follows SAS congruence rule for the given parameters.
Construction of a triangle when two angles of a triangle and a side are known and understanding the side given can only be constructed between the two angles to form a unique triangle. Construction follows ASA congruence rule.
Right angle is one of the angles of the triangle the and assimilating other parameters that are required to complete construction. Construction of a triangle based on RH congruence rule.
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. Construction of a triangle with given parameters sum of two sides and an angle follows SAS congruence rule.
Difference of two sides and an angle are parameters with which a triangle construction is possible, this construction of triangle follows SAS congruence rule.
Constructing a unique triangle with perimeter and with other two parameters the base angles of the triangle follows construction on a triangle using SAS congruence rule.
Concept #: Concurrency in triangles
A set of lines are said to be concurrent if they all intersect at the same point. In the figure below, the three lines are concurrent because they all intersect at a single point P. The point P is called the "point of concurrency". This concept appears in the various centers of a triangle.
All non-parallel lines are concurrent.
Rays and line segments may, or may not be concurrent, even when not parallel.
In a triangle, the following sets of lines are concurrent:
The three medians.
The three altitudes.
The perpendicular bisectors of each of the three sides of a triangle.
The three angle bisectors of each angle in the triangle.
The medians, altitudes, perpendicular bisectors and angle bisectors of a triangle are all concurrent lines. Their point of intersections are called centroid, orthocentre, circumcentre and incentre respectively. Concurrent lines are especially important in triangle geometry, as the three-sided nature of a triangle means there are several special examples of concurrent lines, including the centroid, the circumcenter and the orthocenter.
These points of concurrencies, orthocenter, centroid, and circumcenter of any triangle are collinear that is they lie on the same straight line called the Euler line.
Interactive activity to introduce concurrent lines using examples from our surroundings.
Concept #: Concurrency of medians in triangles.
Median of a triangle is a line segment from a vertex to the midpoint of the opposite side. A triangle has three medians. Each median divides the triangle into two smaller triangles of equal area. The medians of a triangle are concurrent and the point of concurrence is called the centroid. The centroid is always inside the triangle. The centroid is exactly two-thirds the way along each median. i.e the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.
From Latin: centrum - "center", and Greek: -oid -"like" The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter. Refer to the figure . Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip. Once you have found the point at which it will balance, that is the centroid. In the diagram, the medians of the triangle are shown as dotted blue lines.
The centroid of a triangle is where the three medians intersect. This activity will show you how to find the centroid and you’ll explore several geometric relationships related to centroid and medians.
Concept #: Concurrency of altitudes in triangles
The distance between a vertex of a triangle and the opposite side is called the altitude of the triangle. Altitude also refers to the length of the segment. Altitudes can be used to compute the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. A triangle has 3 altitudes. The intersecting point of 3 altitudes of a triangle is known as orthocentre of the triangle. This point may be inside, outside, or on the triangle. If the triangle is obtuse, it will be outside. If the triangle is acute, the orthocentre is inside the triangle. The orthocenter on a right triangle would be directly on the 90° vertex. From Greek: orthos - "straight, true, correct, regular" The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.
An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side. A triangle can have three altitudes. Point of intersection of these lines for different types of triangles is explored.
Concept #: Concurrency of Perpendicular bisectors in a triangle
The perpendicular bisector of a triangle is the perpendicular drawn to a line segment which divides it into two equal parts. The point where the three perpendicular bisectors of a triangle meet is called the circumcentre of a triangle. The circumcentre of a triangle is equidistant from all the three sides vertices of the triangle. This common distance is the crcumradius. The circumcentre is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices. The circumcentre of a triangle lies inside or on a side or outside the triangle according as the triangle is acute or right angled or obtuse. The circumcentre of a right angled triangle is the mid-point of its hypotenuse. Latin: circum - "around" centrum - "center"
One consequence of the Perpendicular Bisector Theorem is that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
Circumcentre for different types of triangles is investigated with this activity and this further explores several geometric relationships related to the circumcentre and perpendicular bisectors.
Concept #: Concurrency of angle bisectors in triangles.
The ray which bisects an angle is called the angle bisector of a triangle. The point of concurrence of angle bisectors of a triangle is called as incentre of the triangle. The incentre always lies inside the triangle. The distance from incentre to all the sides are equal and is referred to as inradius. The circle drawn with inradius is called incircle and touches all sides of the triangle.
If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.
Two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection. If the objects also have the same size, they are congruent. Two triangles are said to be congruent to one another only if their corresponding sides and angles are equal to one another
5. The below video gives information about Angle Sum Property by Gireesh K S and Suchetha S S
OER
Web resources:
Wikipedia, the free encyclopedia : The website gives a comprehensive information on triangles from basics to in depth understanding of the topic.
cbsemathstudy.blogspot.com : The website contains worksheets that can be downloaded. Worksheets for other chapters can also be searched for which are listed based on class.
Bright hub education : The website describes a lesson plan for introducing triangles and lists classroom problems at the end of the lesson that can be solved for better understanding.
CPALMS : The website contains lesson plan, activities and worksheets associated with triangles.
UrbanPro : This website gives downloadable worksheets with problems on triangle construction.
AglaSem Schools : This website lists important questions for math constructions.
National Institute of Open Schooling: This website has good reference notes on concurrency of lines in a triangle for both students and teachers.
Math Open Reference : This website gives activities that can be tried and manipulated online for topics on geometry.
Books and journals
Textbooks:
Karnataka Govt Text book – Class 8 : Part 1 , Part 2
Syllabus documents (CBSE, ICSE, IGCSE etc)
Projects (can include math lab/ science lab/ language lab)
Laboratory Manuals - Mathematics : Click here to refer activity 15,16,18 and 20 which explains the properties of Triangle.
