Sales Toll Free No: 1-855-666-7446

# Triangles

Top
 Sub Topics Triangles are three sided polygons or polygons with the least number of sides. A triangle encloses an area with minimum number of boundaries, which are straight line segments. There are a number of theorems related to triangles. These theorems and properties of triangles are liberally used in proving concepts and solving problems in geometry.

## Triangle Definition

Triangle is a closed figure formed by three sides and three angles.

Triangle ABC shown above is formed by the three sides (line segments) AB, BC and CA. The three angles A, B and C are formed at the pairwise intersections of these sides. The point where two sides of a triangle intersect or an angle is formed is called a vertex.
A triangle is commonly represented by the symbol $\triangle$ followed by the vertices as $\triangle$ABC.

## Types of Triangles

Triangles are classified both by the angles and by the sides.

Types of Triangles by Angles:

 Acute Triangle:All angles in an acute triangle are acute angles. That is, each of them measure less than 90Âº. In $\triangle$ ABC, all the three angles A, B and C measure less than 90 degrees. Obtuse Triangle:In an obtuse triangle, one angle is obtuse i.e. measure of angle is more than 90Âº. In $\triangle$ABC, angle A measures more than90 degrees. Right Triangle:One of the angles of the right triangle is a right angle, that is its measure is exactly 90Âº. In $\triangle$ABC, angle C is a right angle

Types of Triangles by Sides:

 Scalene Triangle:No two sides are congruent in ascalene triangle. All the three sideshave different lengths. Isosceles Triangle:An isosceles triangle has at least twocongruent sides. Equilateral Triangle:All the three sides of an equilateraltriangle are congruent.

Double classification of triangle is also often done, like scalene acute triangle, Right isosceles triangle etc.

## Properties of Triangles

Given below are some of the properties of triangles.
1. The sum of the measures of the three angles of a triangle is equal to 180 degrees. (Angle sum Property)
2. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. (Exterior Angle Theorem).
3. The greater angle has longer side opposite to it.
4. Sum of lengths of any two sides of a triangle is greater than the length of the third side. (Triangle inequality theorem).
5. The line segment joining the mid points of two sides of a triangle is parallel to the third side of the triangle and is half its length (Mid Segment Theorem).
6. The three medians of a triangle intersect at a point and the point of concurrence is called the Centroid of the triangle.
7. The perpendicular bisectors of the sides of a triangle concur at a point which is called the Circumcenter of the triangle.
8. The angle bisectors of a triangle concur at a point called the Incenter of the triangle.
9. The altitudes of a triangle intersect at point called the Orthocenter of the triangle.

## Special Triangles

The right triangles with angle measures 45 - 45 - 90 and 30 - 60 - 90 are called the special right triangles, because of the relationship between the lengths of their sides.

The lengths of sides opposite to angles 45 - 45 - 90 in the special right triangle are in the ratio 1 : 1 : $\sqrt{2}$. In other words, in a 45 - 45 - 90 triangle, the length of the hypotenuse = $\sqrt{2}$ times the length of each leg. This is an Isosceles right triangle as the legs are congruent.

The lengths of sides opposite to angles 30 - 60 - 90 in the special right triangle are in the ratio 1 : $\sqrt{3}$ : 2.
In other words, for the above triangle,
The length of the longer leg is $\sqrt{3}$ times the length of the shorter leg.
The length of the hypotenuse is twice the length of the shorter leg.

## Triangle Congruence

The congruence theorems for triangles are as follows:

Side - Side - Side Congruence: (SSS criterion for congruence)
If the sides of one triangle are congruent to the sides of a second triangle, then the two triangles are congruent.

Side - Angle - Side Congruence: (SAS criterion)
If two sides and included angle are congruent to two sides and included angle of another triangle, then the two triangles are congruent.

Angle - Side - Angle Congruence: (ASA Criterion)
If two angles and included side of one triangle are congruent to two sides and included side of another triangle, then the two triangles are congruent.

Angle - Angle - Side Congruence: (AAS Criterion)
If two angles and a non included side of a triangle are correspondingly congruent to two angles and non included side of another triangle, then the two triangles are congruent.

Hypotenuse - Leg Congruence: (HL Postulate)
If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent.

Leg - Leg Congruence: (LL Theorem)
If the legs of one right triangle are correspondingly congruent to the legs of another right triangle, then the triangles are congruent.

## Perimeter of a Triangle

Perimeter of a triangle is the length measured around it. In other words, it is the sum of the lengths of the sides. If the lengths of the sides are represented by the letters a, b and c, then
Perimeter of the triangle = (a + b + c) units.
Perimeter being a length, has the same unit as the lengths of the sides.

For example, suppose the sides of a triangle are of lengths 7.5 cm, 8 cm and 10.5 cm.
Then, the Perimeter of the triangle = 7.5 + 8 + 10.5 = 26

## Area of a Triangle

The formula used to find the area of a triangle is as follows:
Area of triangle = $\frac{1}{2}$ x base x height = $\frac{1}{2}$ bh
where, base is the length of any side and the height is the length of the altitude on it.

Heron's formula provides a method to calculate the area, when the lengths of the sides are known.
Area of triangle = $\sqrt{s(s - a)(s - b)(s - c)}$
where, a, b and c are the lengths of the sides and s = $\frac{a + b + c}{2}$.
s is called the semi perimeter of the triangle.

## Triangle Problems

Given below are some of the problems on triangles.

### Solved Examples

Question 1: Find the perimeter and area of the triangle whose sides are of length 8 cm, 10 cm and 12 cm.
Solution:
If we take a = 8, b = 10 and c = 12, then
Perimeter of the triangle = a + b + c = 8 + 10 + 12 = 30 cm.
As the lengths of the sides are known, Heron's formula can be used to compute the area of the triangle.
s = $\frac{a + b + c}{2}$$\frac{30}{2}$ = 15
Area of triangle = $\sqrt{s(s - a)(s - b)(s - c)}$
= $\sqrt{15(15 - 8)(15 - 10)(15 - 12)}$
= $\sqrt{1575}$ = 39.69

Area of the triangle is 39.69 sq.cm.

Question 2: The diagonal of a square divides the square into two congruent right isosceles triangles. Find the length of the diagonal, if the length of the side of the square is 6 inches.
Solution:

Solution: The diagonal divides the square into two 45 - 45 - 90 special right triangle and the hypotenuse for both the triangles.
Length of the hypotenuse in a 45 - 45 - 90 triangle = $\sqrt{2}$ times the length of the leg.
Length of the diagonal = $6 \sqrt{2}$ inches.