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While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
  
= Introduction =
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===Concept Map ===
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[[File:Triangles.mm]]
The following is a background literature for
 
teachers. It summarises the things to be known to a teacher to teach
 
this topic more effectively . This literature is meant to be a ready
 
reference for the teacher to develop the concepts, inculcate
 
necessary skills, and impart knowledge in Triangles from Class 6 to
 
Class 10. The idea is to know that a triangle is one of the basic
 
shapes of geometry and how a triangle is formed. To understand the
 
type of Triangles based on their sides and angles. To understand how
 
a exterior angle is formed and the meaning of interior angles of a
 
triangle. To know when two triangles become congruent to each other
 
by understanding the postulates. Meaning of similarity and theorems
 
based on similarity of triangles. Costruction of triangles and to
 
find the Area of the Triangle.
 
  
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===Additional Resources===
= Concept Map =
 
 
[[Image:KOER%20Triangles_html_219b0871.jpg]]
 
 
= TRIANGLES =
 
 
  
Mark three
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==== Resource Title and description ====
non-collinear point P, Q and R on a paper. Join these pints in
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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.,
allpossible ways. The segments are PQ, QR and RP. A simple close
 
curve formed by these three segments is called a triangle. It is
 
named in one of the following ways.
 
 
Triangle PQR or
 
Triangle PRQ or Triangle QRP or Triangle RPQ or Triangle RQP .
 
  
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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.
[[Image:KOER%20Triangles_html_m55b3a2cf.png|picture of equilateral triangle]] PQR
 
 
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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3. [http://www.mathopenref.com/tocs/congruencetoc.html Congruence]
  
In fact, it is the
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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]
polygon with the least number of sides.
 
  
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5. The below video gives information about Angle Sum Property by Gireesh K S and Suchetha S S
  
A triangle PQR
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{{#widget:YouTube|id=GfP8M5GwcdQ}}
consists of all the points on the line segment PQ,QR and RP.
 
  
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====OER====
The three line segments, PQ, QR and RP that form
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* Web resources:
the triangle PQ, are called the sides of the triangle PQR.
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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.
 +
** [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
'''Angles: '''
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* Textbooks
 +
** NCERT Textbooks – [http://ncert.nic.in/textbook/textbook.htm?iemh1=7-15 Class 9]
  
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* Syllabus documents
A triangle has three angles. In figure, the three
 
angles are ∠PQR ∠QRP and ∠RPQ
 
  
   
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====Non-OER====
'''Parts of triangle:'''
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* Web resources:
 +
** [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.
 +
** [http://www.cpalms.org/Public/PreviewResourceLesson/Preview/40261 CPALMS] : The website contains lesson plan, activities and worksheets associated with triangles.
 +
** [https://www.urbanpro.com/cbse-class-9-maths-construction UrbanPro] : This website gives downloadable worksheets with problems on triangle construction.
 +
** [https://schools.aglasem.com/59755 AglaSem Schools] : This website lists important questions for math constructions.
 +
** [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.
 +
** [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
A triangle has six parts, namely, three sides,PQ
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* Textbooks:
QRand RP.Three angles ∠PQR ∠QRP and ∠RPQ. These are also known
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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]
as the elements of a triangle.
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* Syllabus documents (CBSE, ICSE, IGCSE etc)
  
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===Learning Objectives===
'''Vertices of a Triangle '''
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* 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
  
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===Teaching Outlines===
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 thevertex. 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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==== 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
== Evaluation ==
 
 
== Self-Evaluation ==
 
 
== Further    Explorations ==
 
 
# [[http://www.mathsisfun.com/triangle.html]] All about Triangles
 
 
== Enrichment    Activities ==
 
 
 
== Activities ==
 
 
=== Activity 1: Identifying and Naming Triangles ===
 
 
==== Learning Objectives ====
 
 
Identify and name the triangles
 
  
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===== Activities # =====
  
==== Material    and Resources Required ====
+
====== [[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. 
==== Pre-requisites/Instructions ====
 
 
Identify and name the triangles in
 
the following Figure.
 
  
 
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====== [[Measures in triangle|Elements and measures in triangle]] ======
[[Image:KOER%20Triangles_html_m27d3a9c5.png]]
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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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====== [[Interior and exterior angles in triangle]] ======
==== Evaluation ====
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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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[[Category:Triangles]]
# Is it possible to construct a triangle with 3 collinear points?
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[[Category:Class 10]]
# Is it possible to construct a triangle whose sides are 3cm, 4cm and 9cm. Give reason.
 
  
= Classification of Triangles =
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==== Concept #: Types of triangles based on sides and angles ====
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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.
Triangles can be
 
classified in two groups:
 
  
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===== Activities # =====
Triangles
 
differentiated on the basis of their sides.
 
  
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====== [[Types of triangles based on sides]] ======
== Equilateral Triangles: ==
+
A triangle can be drawn with different measures of sides and these sides determine the kind of triangle formed.
 
[[Image:KOER%20Triangles_html_m55b3a2cf.png|picture of equilateral triangle]]
 
A triangle with all sides equal to one another is called an
 
equilateral triangle.
 
  
 +
====== [[Types of triangles based on angles]] ======
 +
A triangle can be drawn with different measures of angles which also determine the kind of triangle formed.
  
== Isosceles Triangle: ==
+
==== Concept #: Theorems and properties ====
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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.  
A triangle with a pair
 
of equal sides is called an isosceles triangle.
 
  
+
===== Activities #  =====
[[Image:KOER%20Triangles_html_m34248090.png]]
 
  
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====== [[Angle sum property]] ======
== Scalene Triangle: ==
+
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.
 
A triangle in which
 
all the sides are of different lengths and no two sides are equal,
 
the triangle is called a scalene triangle.
 
  
 
+
====== [[Angle sum property of a Triangle]] ======
== Triangles differentiated on the basis of their angles. ==
 
 
'''Acute angled triangle.'''
 
  
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====== [[Relation between interior and exterior angles in triangle|Exterior angle theorem]] ======
A triangle whose all
+
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.
angles are acute is called an acute-angled triangle or simply an
 
acute triangle.
 
  
 +
====== [[Exterior angle property of a Triangle]] ======
  
[[Image:KOER%20Triangles_html_m62898a77.png]]
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==== Concept #: Construction of triangles ====
 +
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
  
'''Right Triangles : '''
+
===== Activities # =====
  
+
====== [[Construction of a triangle with 3 sides|Construction of a triangle with three sides]] ======
[[Image:KOER%20Triangles_html_m732d9c3d.png]]A
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Investigating formation of a unique triangle with the given parameters as the three sides. Constriction follows SSS congruence rule.
right triangle has one angle 90°
 
  
       
+
====== [[Construction of a triangle with two sides and included angle|Construction of a triangle with two sides and an angle]] ======
{| border="1"
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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.
|-
 
|
 
  
 +
====== [[Construction of a triangle with two angles and included side]] ======
 +
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.
  
+
====== [[Construction of a right triangle with a side and hypotenuse|Construction of a right angled triangle]] ======
|
+
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.
  
 +
====== [[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]]  ======
 +
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.
  
   
+
====== [[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]] ======
|} 
+
Difference of two sides and an angle are parameters with which a triangle construction is possible, this construction of triangle follows SAS congruence rule.
'''Right Isosceles Triangle :'''
 
  
+
====== [[Construction of a triangle with perimeter and base angles]] ======
[[Image:KOER%20Triangles_html_m37b6213c.png]]Has
+
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.
a right angle (90°), and also two equal angles
 
Can you guess what
 
the equal angles are?
 
  
+
==== 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.
  
 +
[[File:concurrent lines.jpeg|100px|link=http://karnatakaeducation.org.in/KOER/en/index.php/File:Concurrent_lines.jpeg]]
  
 +
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 Obtuse Triangle :'''
 
  
+
The perpendicular bisectors of each of the three sides of a triangle.
[[Image:KOER%20Triangles_html_15a07649.png|Obtuse Triangle]]The
 
Obtuse Triangle has an obtuse angle (an obtuse angle has more than
 
90°). In the picture on the left, the shaded angle is the obtuse
 
angle that distinguishes this triangle
 
  
+
The three angle bisectors of each angle in the triangle.
Since the total degrees in any triangle is 180°,
 
an obtuse triangle can only have one angle that measures more than
 
90°.
 
  
   
+
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.
 +
======Activities #======
 +
======[[Exploring concurrent lines from given surroundings]]======
 +
Interactive activity to introduce concurrent lines using examples from our surroundings.
 +
{| style="height:10px; float:right; align:center;"
 +
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 +
|}
 +
=====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 - &quot;center&quot;, and Greek: -oid -&quot;like&quot;  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.
  
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[[Image:KOER%20Triangles_html_m404a4c0b.gif|link=]]
 
+
======Activities #======
 
+
======[[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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{| style="height:10px; float:right; align:center;"
 
+
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
+
|}
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======[[Medians and centroid of a triangle]]======
== Interior angles of a triangle ==
+
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 interior angles are
+
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 - &quot;straight, true, correct, regular&quot; The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.
those on the inside of the triangle.
+
======Activities #======
 
+
======[[Altitudes and orthocenter of a triangle]]======
+
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"
 
[[Image:KOER%20Triangles_html_m55b3a2cf.png]]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
== Exterior Angles of a triangle ==
 
 
An exterior angle is formed by extending any side
 
of the triangle.
 
 
 
 
 
 
 
 
 
 
 
[[Image:KOER%20Triangles_html_9fb9ff3.png]]
 
 
 
 
 
 
== Summary of triangle centres ==
 
 
There are many types of
 
triangle centers. Below are four of the most common.
 
 
 
                       
 
{| border="1"
 
|-
 
|
 
Incenter
 
 
 
 
|
 
[[Image:KOER%20Triangles_html_ef07362.gif]]
 
 
 
 
|
 
Located at intersection of the angle bisectors.
 
See Triangle
 
incenter definition
 
 
 
 
|-
 
|
 
Circumcenter
 
 
 
 
|
 
[[Image:KOER%20Triangles_html_68b61322.gif]]
 
 
 
 
|
 
Located at intersection of the perpendicular bisectors of the
 
sides.
 
See Triangle circumcenter definition
 
 
 
 
|-
 
|
 
Centroid
 
 
 
 
|  
 
[[Image:KOER%20Triangles_html_16723946.gif]]
 
 
 
 
|
 
Located at intersection of medians.
 
See Centroid of a
 
triangle
 
 
 
 
|-
 
|
 
Orthocenter
 
 
 
 
|
 
[[Image:KOER%20Triangles_html_7aa50a01.gif]]
 
 
 
 
|
 
Located at intersection of the altitudes of the triangle.
 
See
 
Orthocenter of a triangle
 
 
 
 
|}
 
In the case of an equilateral triangle, all four
 
of the above centers occur at the same point.
 
 
 
 
The Incenter of a
 
triangle
 
 
 
 
Latin: in - &quot;inside,
 
within&quot; centrum - &quot;center&quot;
 
 
 
 
The point where the
 
three angle bisectors of a triangle meet.
 
One of a triangle's
 
points of concurrency.
 
 
 
 
Try this Drag the
 
orange dots on each vertex to reshape the triangle. Note the way the
 
three angle bisectors always meet at the incenter.
 
 
 
 
 
 
 
 
 
 
 
One of several centers the triangle can have, the
 
incenter is the point where the angle bisectors intersect. The
 
incenter is also the center of the triangle's incircle - the largest
 
circle that will fit inside the triangle.
 
 
 
 
== Centroid of a Triangle ==
 
 
From Latin: centrum -
 
&quot;center&quot;, and Greek: -oid -&quot;like&quot; The point where
 
the three medians of the triangle intersect. The 'center of gravity'
 
of the triangle
 
One of a triangle's points of concurrency. Try
 
this Drag the orange dots at A,B or C and note where the centroid is
 
for various triangle shapes.
 
 
 
 
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.
 
 
 
 
[[Image:KOER%20Triangles_html_m404a4c0b.gif]]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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.
 
 
 
 
A fascinating fact is that the centroid is the
 
point where the triangle's medians intersect. See medians of a
 
triangle for more information. In the diagram above, the medians of
 
the triangle are shown as dotted blue lines.
 
 
 
 
'''Centroid facts'''
 
 
 
 
* The centroid is always inside the triangle
 
* Each median divides the triangle into two smaller triangles of equal area.
 
* The centroid is exactly two-thirds the way along each median.
 
Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex. These lengths are shown on the one of the medians in the figure at the top of the page so you can verify this property for yourself.
 
 
== Orthocenter of a Triangle ==
 
 
From Greek: orthos -
 
&quot;straight, true, correct, regular&quot; The point where the
 
three altitudes of a triangle intersect. One of a triangle's points
 
of concurrency.
 
 
 
 
Try this Drag the
 
orange dots on any vertex to reshape the triangle. Notice the
 
location of the orthocenter.
 
 
 
 
 
 
 
 
 
The altitude of a triangle (in the sense it used
 
here) is a line which passes through a vertex of the triangle and is
 
perpendicular to the opposite side. There are therefore three
 
altitudes possible, one from each vertex. See Altitude definition.
 
 
 
   
 
It turns out that all three altitudes always
 
intersect at the same point - the so-called orthocenter of the
 
triangle.
 
 
 
 
The orthocenter is not always inside the triangle.
 
If the triangle is obtuse, it will be outside. To make this happen
 
the altitude lines have to be extended so they cross. Adjust the
 
figure above and create a triangle where the orthocenter is outside
 
the triangle. Follow each line and convince yourself that the three
 
altitudes, when extended the right way, do in fact intersect at the
 
orthocenter.
 
 
 
 
=== Activity 1 Types of Triangles ===
 
 
 
 
 
 
 
 
 
==== Learning Objectives ====
 
 
Be able to identify triangles.
 
  
+
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.
==== Material and Resources Required ====
+
======Activities #======
+
[[Perpendicular bisectors and circumcenter of a triangle|'''Perpendicular bisectors and circumcenter of a triangle''']]
==== Pre-requisites/Instructions ====
 
 
Identify the types of triangles.
 
  
+
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.
 +
======Activities #======
 +
======[[Angular bisectors and incenter of a triangle]]======
 +
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.
  
+
==== Concept #: [[Similar and congruent triangles]] ====
 +
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
  
 +
==== Solved problems/ key questions (earlier was hints for problems).[edit | edit source] ====
  
         
+
=== Projects (can include math lab/ science lab/ language lab)[edit | edit source] ===
{| border="1"
 
|-
 
|
 
[[Image:KOER%20Triangles_html_m7d1c3f58.png|Equilateral Triangle]]
 
 
 
 
|
 
[[Image:KOER%20Triangles_html_28f84d5e.png|Isosceles triangle]]
 
 
 
 
|
 
[[Image:KOER%20Triangles_html_36280b0d.png|Scalene triangle]]
 
 
 
 
|} 
 
 
 
 
 
 
 
 
 
 
         
 
{| border="1"
 
|-
 
|
 
[[Image:KOER%20Triangles_html_21013b99.png|Right triangle]]
 
 
 
 
|
 
[[Image:KOER%20Triangles_html_4b875456.png|Obtuse triangle]]
 
 
 
 
|
 
[[Image:KOER%20Triangles_html_5c56ee05.png|Acute triangle]]
 
 
 
 
|} 
 
 
 
 
 
 
==== Evaluation ====
 
 
# Can a scalene triangle also be a right-angled triangle ? If yes can you draw one ?
 
 
=== Activity 2 Similar Triangles ===
 
 
'''Learning Objective'''
 
 
 
 
To show similar planar
 
figures, discuss congruence and properties of congruent/ similar
 
triangles
 
 
 
 
 
 
 
 
 
==== Material and Resources Required ====
 
 
Blackboard
 
 
 
 
Geogebra files +
 
projector
 
 
 
 
Calculator
 
 
 
 
 
 
'''[[Pre-requisites/Instructions]] '''
 
 
 
 
* Planar figures and triangles
 
* Draw pairs of figures on the board [ both similar and dissimilar]; they can identify overlap of congruent figures
 
* Ask the children to identify
 
* If the children know the names of the theorem, ask them to explain- ask them what is SSS, AAA, ASA
 
* Show ratio and give the idea of proportionality
 
* Geogebra files. When I change the sides/ proportion, the triangles change in size. But the proportion remains the same, angle remains the same
 
* With calculator they verify proportion (this is very very useful for involving the whole class) they all can see the proportion remains constant though the size changes
 
* Show the arithmetic behind the proportion
 
 
==== Evaluation ====
 
 
[Activity evaluation -
 
What should the teacher watch for when you do the activity; based on
 
what they know change]
 
 
 
 
* Confusion between congruence and similarity
 
* When they give the theorem, if they cannot identify included side and angle
 
* When there is a wrong answer, identify what is the source of the confusion – sides, ratio and proportion
 
 
* Direct substitution
 
 
 
 
 
 
 
== Evaluation ==
 
 
== Self-Evaluation ==
 
 
== Further Explorations ==
 
 
== Enrichment Activities ==
 
 
= Pythagorean Theorem =
 
 
 
 
 
 
         
 
{| border="1"
 
|-
 
|
 
Pythagoras' Theorem was 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:
 
 
 
 
[[Image:KOER%20Triangles_html_m2f096af4.png]]
 
 
 
 
 
[[Image:KOER%20Triangles_html_4bd439df.png]]
 
 
 
 
|}
 
 
 
 
 
                   
 
{| border="1"
 
|-
 
|
 
 
 
 
 
 
|-
 
|
 
 
 
 
 
 
|-
 
|
 
 
 
 
 
 
|-
 
|
 
 
 
 
 
 
|
 
 
 
 
 
 
|-
 
|
 
 
 
 
 
 
|}                     
 
{| border="1"
 
|-
 
|
 
'''Pythagoras' Theorem in
 
Three Dimensions'''
 
 
 
 
A
 
three-dimensional object can be described by three measurements -
 
length, width and height.
 
 
 
 
|-
 
|
 
[[Image:KOER%20Triangles_html_679fe2f6.png]][[Image:KOER%20Triangles_html_m570261d2.png]]
 
 
 
 
|-
 
|
 
We
 
can use Pythagoras' Theorem to find the length of the longest
 
straw that will fit inside
 
 
 
 
the
 
box or cylinder.
 
 
 
 
|-
 
|
 
== Evaluation ==
 
 
== Self-Evaluation ==
 
 
== Further    Explorations ==
 
 
 
== Enrichment    Activities ==
 
 
|-
 
|
 
= See    Also =
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
= Teachers    Corner =
 
 
Sucheta SS doing
 
a lesson on similar triangles using GeoGebra in the classroom
 
 
 
 
[[Image:KOER%20Triangles_html_m3d25043b.jpg|600px]]
 
 
 
 
 
 
= Books =
 
 
 
 
 
 
 
 
 
= References =
 
 
 
|-
 
|
 
 
 
 
 
 
|}
 
<gallery>
 
 
 
Image:KOER Triangles_html_4b875456.png
 
Image:KOER Triangles_html_4bd439df.png
 
Image:KOER Triangles_html_5c56ee05.png
 
Image:KOER Triangles_html_7aa50a01.gif
 
Image:KOER Triangles_html_9fb9ff3.png
 
Image:KOER Triangles_html_15a07649.png
 
Image:KOER Triangles_html_28f84d5e.png
 
Image:KOER Triangles_html_68b61322.gif
 
Image:KOER Triangles_html_219b0871.jpg
 
Image:KOER Triangles_html_679fe2f6.png
 
Image:KOER Triangles_html_21013b99.png
 
Image:KOER Triangles_html_36280b0d.png
 
Image:KOER Triangles_html_16723946.gif
 
Image:KOER Triangles_html_64378236.png
 
Image:KOER Triangles_html_ef07362.gif
 
Image:KOER Triangles_html_m2f096af4.png
 
Image:KOER Triangles_html_m3d25043b.jpg
 
Image:KOER Triangles_html_m7d1c3f58.png
 
Image:KOER Triangles_html_m27d3a9c5.png
 
Image:KOER Triangles_html_m37b6213c.png
 
Image:KOER Triangles_html_m55b3a2cf.png
 
Image:KOER Triangles_html_m404a4c0b.gif
 
Image:KOER Triangles_html_m732d9c3d.png
 
Image:KOER Triangles_html_m62898a77.png
 
Image:KOER Triangles_html_m570261d2.png
 
Image:KOER Triangles_html_m34248090.png
 
  
</gallery>
+
=== Assessments - question banks, formative assessment activities and summative assessment activities ===

Latest revision as of 14:14, 19 December 2020

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Contents

Concept Map

[maximize]

Additional Resources

Resource Title and description

1.Triangles- This resource contain all information related to triangle like definition, types of triangle, perimeter, congruency etc.,

2. Triangles- This helps to know the aplication of geometry in our daily life, it contain videos and interactives.

3. Congruence

4. Similarity and Congruence

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.
  • Books and journals
  • Textbooks
  • Syllabus documents

Non-OER

  • Web resources:
    • 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:
  • Syllabus documents (CBSE, ICSE, IGCSE etc)

Learning Objectives

  • 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

Activities #
Formation of a triangle

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. 

Elements and measures in triangle

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 and exterior angles in triangle

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.

Activities #
Types of triangles based on sides

A triangle can be drawn with different measures of sides and these sides determine the kind of triangle formed.

Types of triangles based on angles

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.

Activities #
Angle sum property

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.

Angle sum property of a Triangle
Exterior angle theorem

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.

Exterior angle property of a Triangle

Concept #: Construction of triangles

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

Activities #
Construction of a triangle with three sides

Investigating formation of a unique triangle with the given parameters as the three sides. Constriction follows SSS congruence rule.

Construction of a triangle with two sides and an angle

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 with two angles and included side

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.

Construction of a right angled triangle

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.

Construction of a triangle with a side, an angle and sum of two sides

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.

Construction of a triangle with a side, an angle and difference to two sides

Difference of two sides and an angle are parameters with which a triangle construction is possible, this construction of triangle follows SAS congruence rule.

Construction of a triangle with perimeter and base angles

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.

Concurrent lines.jpeg

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.

Activities #
Exploring concurrent lines from given surroundings

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.

KOER Triangles html m404a4c0b.gif

Activities #
Marking centroid of the triangle

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.

Medians and centroid of a triangle

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.

Activities #
Altitudes and orthocenter of a triangle

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.

Activities #

Perpendicular bisectors and circumcenter of a 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.

Activities #
Angular bisectors and incenter of a triangle

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.

Concept #: Similar and congruent triangles

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

Solved problems/ key questions (earlier was hints for problems).[edit | edit source]

Projects (can include math lab/ science lab/ language lab)[edit | edit source]

Assessments - question banks, formative assessment activities and summative assessment activities