Revision Notes on The Triangle and its Properties

Learning Goal

1. Triangles- (closed curve figure made up of three-line segments)
2. Triangles based on sides (Equilateral, Isosceles, Scalene)
3. Triangles based on angles (Acute, Right, Obtuse)
4. Medians of a Triangle (Joining vertex and the midpoint of its opposite side)
5. Altitudes of a Triangle (Joining the vertex and the perpendicular to the opposite side)
6. Centroid (Intersection of medians)
7. Orthocenter (Intersection of altitudes)
8. The Exterior Angle of a Triangle (extending any side of the triangle)
9. Exterior Angle Property of the Triangle (E.A. = Sum of two opposite interior angles)
10. Angle Sum Property of a Triangle (sum = 180⁰)
11. Sum of the length of the two sides of a triangle (a + b > c)
12. Pythagoras theorem (h² = b² + p²)

Triangle

A closed curve figure made up of three-line segments is called triangle.

It has three vertices, sides and angles.

Here, in ∆ABC,

AB, BC and CA are the three sides.

A, B and C are three vertices.

∠A, ∠B and ∠C are the three angles.

Types of Triangles on the basis of sides

Types of Triangles on the basis of angles

Medians of a Triangle

The line segment which made by joining any vertex and the midpoint of its opposite side is called median.

Median divides the side into two equal parts.

Every triangle has three medians like AE, CD and BF in the above triangle.

The point where all the three medians intersect each other is called Centroid.

Altitudes of a Triangle

The line segment made by joining the vertex and the perpendicular to the opposite side is called altitude.

Altitude is the height if we take the opposite side as the base.

The altitude form angle of 90°.

There are three altitudes possible in a triangle.

The point of intersection of all the three altitudes is called Orthocenter.

The Exterior Angle of a Triangle

The angle made by extending any side of the triangle is called an exterior angle.

An exterior angle must form a linear pair with one of the interior angles of the triangle.

There are only two exterior angles possible at each of the vertices.

Here ∠4 and ∠5 are the exterior angles of the vertex but ∠6 is not the exterior angle as it is not adjacent to any of the interior angles of the triangle.

Exterior Angle Property of the Triangle

Exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Here, ∠d = ∠a + ∠b

This is called the Exterior angle property of a triangle.

Example

Find the value of “x”.

Solution

x is the exterior angle of the triangle and the two given angles are the opposite interior angles.

Hence,

x = 64°+ 45°

x = 109°

Angle Sum Property of a Triangle

The sum of all the interior angles of a triangle is 180°.

Example

Find the value of x and y in the given triangle.

Solution

x + 58° = 180° (linear pair)

x = 180° – 58°

x = 122°

We can find the value of y by two properties-

1. Angle sum property

60° + 58° + y = 180°

y = 180°- (60° + 58)

y = 62°

2. Exterior angle property

x = 60°+ y

122° = 60° + y

y = 122° – 60°

y = 62°

Two Special Triangles

1. Equilateral Triangle

It is a triangle in which all the three sides and angles are equal.

2. Isosceles Triangle

It is a triangle in which two sides are equal and the base angles opposite to the equal sides are also equal.

Sum of the length of the two sides of a triangle

Sum of the length of the two sides of a triangle will always be greater than the third side, whether it is an equilateral, isosceles or scalene triangle.

Example

Check whether it is possible to make a triangle using these measurements or not?

1. 3 cm, 4 cm, 7 cm

We have to check whether the sum of two sides is greater than the third side or not.

4 + 7 = 11

3 + 7 = 10

3 + 4 =7

Here the sum of the two sides is equal to the third side so the triangle is not possible with these measurements.

2. 2 cm, 5 cm, 6 cm

2 + 5 = 7

6 + 5 =11

6 + 2 = 8

Here the sum of the two sides is greater than the third side so the triangle could be made with these measurements.

Right Angled Triangle

A triangle whose one angle is 90°is called right-angled triangle.

The opposite side of 90° is called hypotenuse.

Hypotenuse is the largest leg of the triangle.

The other two sides are called base and perpendicular.

Pythagoras theorem

In a right-angled triangle,

(Hypotenuse)² = (base)² + (height)²

The reverse of Pythagoras theorem is also applicable, i.e. if the Pythagoras property holds in a triangle then it must be a right-angled triangle.

Example

Find the value of x in the given triangle if the hypotenuse is 5 cm and height is 4 cm.

Solution

Given:

Hypotenuse = 5 cm

Height = 4 cm

Base = x cm

(Hypotenuse)² = (base)² + (height)²

5² = x² + 4²

x² = 5² – 4²

x² = 25 – 16

x² = 9

x = 3 cm