# 8.10.2. Converse of Pythagoras Theorem

In the isosceles triangle below, cm, cm, and cm. Is a right-angled triangle?

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### Example

In the isosceles triangle below, cm, cm, and cm. Is a right-angled triangle?

### Solution

Step 1: Find the length of .

First, find the length of . Since is isosceles and , then bisects . Therefore,

Then use Pythagoras's theorem. Substitute the known lengths in , and solve for the length of .

Step 2: Find the length of .

Then use Pythagoras's theorem to find the length of . Substitute the known lengths in , and solve for the length of . Approximate to the nearest hundredth.

Then add to find the length of .

Step 3: Determine if is a right-angled triangle.

The converse of Pythagoras' theorem states that if the square of the length of a side is equal to the sum of the squares of the other two sides, then the angle opposite to this side is a right angle. For ,

Since , then and is not a right-angled triangle.

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