## What is converse of Pythagoras theorem?

In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right

angle

If $AC^2 = AB^2 + BC^2$

then $\angle ABC =90^0$

## Proof of converse of Pythagoras theorem

Let ABC be the triangle and given

$AC^2 = AB^2 + BC^2$

To Prove: $\angle ABC =90^0$

Proof:

Lets us construct a triangle PQR right angled at Q such that PQ = AB and QR = BC

Now in right angle triangle PQR , By Pythagoras theorem

$PR^2 = PQ^2 + QR^2$

or

$PR^2 = AB^2 + BC^2$ ( as PQ = AB and QR = BC)

Therefore

$PR^2 = AC^2$

PR=AC

In triangle ABC and PQR

PQ = AB , QR = BC, PR=AC

By SSS congruence

$\Delta ABC \cong \Delta PQR$

Now from CPCT

$\angle B = \angle Q = 90$

Hence Proved

## Examples

**Question 1**: Given a triangle with sides of length 6 cm, 8 cm, and 10 cm, is the triangle right-angled?

**Answer**:

Check if 10² = 6² + 8².

100 = 36 + 64

100 = 100

Yes, the triangle is right-angled.

**Question 2**: A triangle has sides measuring 7 cm, 24 cm, and 25 cm. Determine whether the triangle is right-angled or not.

**Answer**:

Check if 25² = 7² + 24².

625 = 49 + 576

625 = 625

Yes, the triangle is right-angled.

**Question 3:** Can a triangle with side lengths of 5 cm, 12 cm, and 14 cm be a right-angled triangle? Use the converse of Pythagoras Theorem to justify your answer.

**Answer**:

Check if 14² = 5² + 12².

196 = 25 + 144

196 = 169

No, the triangle is not right-angled.

**Question 4**:

Determine if a triangle with side lengths of 9 cm, 40 cm, and 41 cm is a right-angled triangle.

**Answer**:

Check if 41² = 9² + 40².

1681 = 81 + 1600

1681 = 1681

Yes, the triangle is right-angled.

**Question 5**: A triangle has sides of length 15 cm, 20 cm, and 25 cm. Use the converse of Pythagoras Theorem to check if the triangle is right-angled.

**Answer**: Check if 25² = 15² + 20².

625 = 225 + 400

625 = 625

Yes, the triangle is right-angled.

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