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Line 120: − + − Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is the point of concurrency+ + + + + + + Line 116:
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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'''].
===Concept Map ===
===Concept Map ===
[[File:5. Triangles.mm]]
[[File:Triangles.mm]]
===Additional Resources===
===Additional Resources===
==== Resource Title and description ====
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.,
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.
3. [http://www.mathopenref.com/tocs/congruencetoc.html Congruence]
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]
5. The below video gives information about Angle Sum Property by Gireesh K S and Suchetha S S
{{#widget:YouTube|id=GfP8M5GwcdQ}}
====OER====
====OER====
* Web resources:
* Web resources:
** [https://www.urbanpro.com/cbse-class-9-maths-construction UrbanPro] : This website gives downloadable worksheets with problems on triangle construction.
** [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.
** [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.
* Books and journals
* Books and journals
==== Concept #: Formation of a triangle, elements of a triangle and its measures ====
==== 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 parts on which most other figures depend. Here we will be investigating triangles and related its properties
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 # =====
===== Activities # =====
====== [[Interior and exterior angles in triangle]] ======
====== [[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.
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.
[[Category:Triangles]]
[[Category:Triangles]]
[[Category:Class 8]]
[[Category:Class 10]]
==== Concept #: Types of triangles based on sides and angles ====
==== Concept #: Types of triangles based on sides and 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.
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]] ====
==== 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:
In a triangle, the following sets of lines are concurrent:
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.
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 #=====
======Activities #======
======[[Exploring concurrent lines from given surroundings]]======
======[[Exploring concurrent lines from given surroundings]]======
Interactive activity to introduce concurrent lines using examples from our surroundings.
Interactive activity to introduce concurrent lines using examples from our surroundings.
[[Image:KOER%20Triangles_html_m404a4c0b.gif|link=]]
[[Image:KOER%20Triangles_html_m404a4c0b.gif|link=]]
=====Activities #=====
======Activities #======
======[[Marking centroid of a triangle|Marking centroid of the triangle]]======
======[[Marking centroid of a triangle|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.
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.
=====Concept #: Concurrency of altitudes in triangles=====
=====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.
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 #=====
======Activities #======
======[[Altitudes and orthocenter of a triangle]]======
======[[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.
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.
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.
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 #=====
======Activities #======
[[Perpendicular bisectors and circumcenter of a triangle|'''Perpendicular bisectors and circumcenter of a triangle''']]
[[Perpendicular bisectors and circumcenter of a triangle|'''Perpendicular bisectors and circumcenter of a triangle''']]
If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.
If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.
=====Activities #=====
======Activities #======
======[[Angular bisectors and incenter of a triangle]]======
======[[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.
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.