Home | | Maths 8th Std | Converse of Pythagoras Theorem

# Converse of Pythagoras Theorem

If in a triangle, the square on the greatest side is equal to the sum of squares on the other two sides, then the triangle is right angled triangle.

Converse of Pythagoras Theorem

If in a triangle, the square on the greatest side is equal to the sum of squares on the other two sides, then the triangle is right angled triangle.

Example:

In the triangle ABC,

AB2 + AC2 = 112 + 602 = 3721 = 612 = BC2

Hence, ∆ABC is a right angled triangle. (i) There are special sets of numbers a, b and c that makes the Pythagorean relationship true and these sets of special numbers are called Pythagorean triplets. Example: (3, 4, 5) is a Pythagorean triplet.

(ii) Let k be any positive integer greater than 1 and (a, b, c) be a Pythagorean triplet, then (ka, kb, kc) is also a Pythagorean triplet.

Examples: So, it is clear that we can have infinitely many Pythagorean triplets just by multiplying any Pythagorean triplet by k.

We shall now see a few examples on the use of Pythagoras theorem in problems.

Example 5.11

In the figure, AB AC

a) What type of is ABC?

b) What are AB and AC of the ∆ABC ?

c) What is CB called as ?

d) If AC = AB then, what is the measure of B and C ? Solution:

a) ∆ABC is right angled as AB AC at A.

b) AB and AC are legs of ∆ ABC .

c) CB is called as the hypotenuse.

d) B + C = 90º and equal angles are opposite to equal sides. Hence, B = C =90º /2 =45º

Example 5.12

Can a right triangle have sides that measure 5cm, 12cm and 13cm?

Solution:

Take a = 5 , b = 12 and c = 13

Now, a 2 + b2 = 52 + 122 = 25 + 144 = 169 = 132 = c2

By the converse of Pythagoras theorem, the triangle with given measures is a right angled triangle.

Example 5.13

A 20-feet ladder leans against a wall at height of 16 feet from the ground. How far is the base of the ladder from the wall? Solution:

The ladder, wall and the ground form a right triangle with the ladder as the hypotenuse. From the figure, by Pythagoras theorem,

202 = 162 + x2

400 = 256 + x2

x2 = 400 256 = 144 = 122

x =12 feet

Therefore, the base (foot) of the ladder is 12 feet away from the wall.

Activity

1. We can construct sets of Pythagorean triplets as follows.

Let m and n be any two positive integers (m > n):

(a, b, c) is a Pythagorean triplet if a = m2 n2 , b = 2mn and c = m2 + n2 (Think, why?)

Complete the table. 2. Find all integer-sided right angled triangles with hypotenuse 85.

Solution:

(x + y)2 − 2xy = 852 Pythagorean triplets with hypotenuse 85.

13, 84, 85

36, 77, 85

40, 75, 85

51, 68, 85

Example 5.14

Find LM, MN, LN and also the area of ∆ LON. Solution:

From ∆LMO, by Pythagoras theorem,

LM2 = OL2 OM2

LM2 =132 122 =169 144 = 25 = 52

LM = 5 units

From ∆NMO, by Pythagoras theorem,

MN2 = ON2 OM2

= 152122 = 225144 = 81= 92

MN = 9 units

Hence, LN = LM + MN = 5 + 9 = 14 units

Area of ∆LON = 1/2 × base × height

= 1/2 × LN × OM

= 1/2 × 14 × 12 = 84 square units.

Example 5.15

∆ABC is equilateral and CD of the right triangle BCD is 8 cm.

Find the side of the equilateral ∆ABC and also BD. Solution:

As ∆ABC is equilateral from the figure, AB=BC=AC= (x−2) cm.

From ∆BCD, by Pythagoras theorem

BD2 = BC2 + CD2

( x + 2)2 = ( x 2)2 + 82

x2 + 4x + 4 = x24x + 4 + 82

8 x = 82

x = 8cm

The side of the equilateral ∆ ABC = 6 cm and BD =10 cm.

Tags : Geometry | Chapter 5 | 8th Maths , 8th Maths : Chapter 5 : Geometry
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
8th Maths : Chapter 5 : Geometry : Converse of Pythagoras Theorem | Geometry | Chapter 5 | 8th Maths