Triangles

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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

KOER Triangles html 219b0871.jpg

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 .


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.


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.


Evaluation

Self-Evaluation

Further Explorations

  1. [[1]] All about Triangles

Enrichment Activities

Activities

Activity 1: Identifying and Naming Triangles

Learning Objectives

Identify and name the triangles


Material and Resources Required

Pre-requisites/Instructions

Identify and name the triangles in the following Figure.


KOER Triangles html m27d3a9c5.png


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:

picture of equilateral triangle 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.


KOER Triangles html m34248090.png


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.


KOER Triangles html m62898a77.png


Right Triangles :


KOER Triangles html m732d9c3d.pngA right triangle has one angle 90°




Right Isosceles Triangle :


KOER Triangles html m37b6213c.pngHas a right angle (90°), and also two equal angles Can you guess what the equal angles are?






The Obtuse Triangle :


Obtuse TriangleThe 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.



KOER Triangles html m55b3a2cf.png










Exterior Angles of a triangle

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




KOER Triangles html 9fb9ff3.png


Summary of triangle centres

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


Incenter


KOER Triangles html ef07362.gif


Located at intersection of the angle bisectors. See Triangle incenter definition


Circumcenter


KOER Triangles html 68b61322.gif


Located at intersection of the perpendicular bisectors of the sides. See Triangle circumcenter definition


Centroid


KOER Triangles html 16723946.gif


Located at intersection of medians. See Centroid of a triangle


Orthocenter


KOER Triangles 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 - "inside, within" centrum - "center"


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 - "center", and Greek: -oid -"like" 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.


KOER Triangles 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 - "straight, true, correct, regular" 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.


Material and Resources Required

Pre-requisites/Instructions

Identify the types of triangles.





Equilateral Triangle


Isosceles triangle


Scalene triangle





Right triangle


Obtuse triangle


Acute triangle



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



Evaluation

Self-Evaluation

Further Explorations

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:


KOER Triangles html m2f096af4.png


KOER Triangles html 4bd439df.png









Pythagoras' Theorem in Three Dimensions


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


KOER Triangles html 679fe2f6.pngKOER Triangles 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


KOER Triangles html m3d25043b.jpg


Books

References