Question

In ΔABC,AB = 20 cm, AC = 15 cm. The length of the altitude AN is 12 cm. Prove that ΔABC is a right triangle.

Answer 1

pythagorean theorem converse

Explanation:

Answer 2

Given : In ΔABC,AB = 20 cm, AC = 15 cm and the length of the altitude AN is 12 cm.

To prove: Triangle ABC is a right angled triangle

Proof:

Since AN is an altitude, so it is a line segment through a vertex and perpendicular to a line containing the base.

Consider the triangle ANB,

By Pythagoras theorem,

`(AB)^2 = (AN)^2+(BN)^2`

`(20)^2 = (12)^2+(BN)^2`

`(BN)^2=400-144`

`(BN)^2 = 256`

So, BN = 16 cm.

Consider the triangle ANC,

By Pythagoras theorem,

`(AC)^2 = (AN)^2+(NC)^2`

`(15)^2 = (12)^2+(NC)^2`

`(NC)^2=225-144`

`(NC)^2 = 81`

So, NC = 9 cm.

Therefore, BC = BN + NC = 16 + 9 = 25cm.

Now, consider
`(AB)^2 + (AC)^2=(BC)^2`

`(20)^2 + (15)^2=(25)^2`

225 = 225

Therefore, By converse of Pythagoras theorem.

Triangle ABC is a right angled triangle.