Triangles

= Introduction = 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.

= Concept Map = = TRIANGLES =

Mark three non-collinear point P, Q and R on a paper. Join these pints in 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.

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.

In fact, it is the polygon with the least number of sides.

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.

Angles: 

A triangle has three angles. In figure, the three angles are ∠PQR ∠QRP and ∠RPQ

Parts of triangle:

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.

Vertices of a Triangle 

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.

Further  Explorations

 * 1) [] 				All about Triangles

Learning Objectives
Identify and name the triangles

Pre-requisites/Instructions
Identify and name the triangles in the following Figure.



Evaluation

 * 1) Is it possible 				to construct a triangle with 3 collinear points?
 * 2) Is it possible 				to construct a triangle whose sides are 3cm, 4cm and 9cm. Give 				reason.

= Classification of Triangles = Triangles can be classified in two groups:

Triangles differentiated on the basis of their sides.

Equilateral Triangles:
A triangle with all sides equal to one another is called an equilateral triangle.

Isosceles Triangle:
A triangle with a pair of equal sides is called an isosceles triangle.



Scalene Triangle:
A triangle in which all the sides are of different lengths and no two sides are equal, the triangle is called a scalene triangle.

Triangles differentiated on the basis of their angles.
Acute angled triangle.

A triangle whose all angles are acute is called an acute-angled triangle or simply an acute triangle.



Right Triangles : 

A right triangle has one angle 90°

Right Isosceles Triangle :

Has a right angle (90°), and also two equal angles Can you guess what the equal angles are?

The 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

Since the total degrees in any triangle is 180°, an obtuse triangle can only have one angle that measures more than 90°.

Interior angles of a triangle
The interior angles are those on the inside of the triangle.



Exterior Angles of a triangle
An exterior angle is formed by extending any side of the triangle.



Summary of triangle centres
There are many types of triangle centers. Below are four of the most common.

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.



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

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.
 * 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.

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.

Learning Objectives
Be able to identify triangles.

Pre-requisites/Instructions
Identify the types of triangles.

Evaluation

 * 1) 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

Enrichment Activities
= Pythagorean Theorem =

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:





Pythagoras' Theorem in			Three Dimensions

A three-dimensional object can be described by three measurements - length, width and height.



We can use Pythagoras' Theorem to find the length of the longest straw that will fit inside

the box or cylinder.

Enrichment  Activities
= See   Also =

= Teachers   Corner = Suchetha. S. S Asst. Teacher ( Mathematics ) GJC Thyamagondlu. Nelamangala Talluk Bangalore Rural District doing a lesson on similar triangles using GeoGebra in the classroom

GeoGebra Contributions

 * 1) The GeoGebra file below to understand Similar Triangles
 * 2) Similar Triangles Part 1 http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_1.html
 * 3) Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_1.ggb
 * 4) Similar Triangles Part 2 http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_2.html
 * 5) Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_2.ggb
 * 6) Similar Triangles Part 3 http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_3.html
 * 7) Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/Similar_Triangles_3.ggb
 * 8) See a video to understand this concept http://www.youtube.com/watch?v=BI-rtfZVXy0


 * 1) The GeoGebra file below verifies the Thales theorem
 * 2) Thales Theorem http://www.karnatakaeducation.org.in/KOER/Maths/thales_theorem.html
 * 3) Download ggb file here http://www.karnatakaeducation.org.in/KOER/Maths/thales_theorem.ggb
 * 4) See a video that proves this theorem http://www.youtube.com/watch?v=Y-6yYsuGLoc

= Books =

= References =