Converse of Pythagorean Theorem

Converse of Pythagorean Theorem states that:

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

Given: A ∆PQR in which PR² = PQ² + QR²

To prove: ∠Q = 90°

Construction: Draw a ∆XYZ such that XY = PQ, YZ = QR and ∠Y = 90°

$Converse of Pythagorean Theorem$

So, by Pythagora’s theorem we get,

XZ² = XY² + YZ²

⇒ XZ² = PQ² + QR² ……….. (i), [since XY = PQ and YZ = QR]

But, PR² = PQ² + QR² ………… (ii), [given]

From (i) and (ii) we get,

PR² = XZ² ⇒ PR = XZ

Now, in ∆PQR and ∆XYZ, we get

PQ = XY,

QR = YZ and

PR = XZ

Therefore ∆PQR ≅ ∆XYZ

Hence ∠Q = ∠Y = 90°

Word problems using the Converse of Pythagorean Theorem:

1. The side of a triangle are of length 4.5 cm, 7.5 cm and 6 cm. Is this triangle a right triangle? If so, which side is the hypotenuse?

Solution:

We know that hypotenuse is the longest side. If 4.5 cm, 7.5 cm and 6 cm are the lengths of angled triangle, then 7.5 cm will be the hypotenuse.

Using the converse of Pythagoras theorem, we get

(7.5)² = (6)² + (4.5)²

⇒ 56.25 = 36 + 20.25

⇒ 56.25 = 56.25

Since, both the sides are equal therefore, 4.5 cm, 7.5 cm and 6 cm are the side of the right angled triangle having hypotenuse 7.5 cm.

2. The side of a triangle are of length 8 cm, 15 cm and 17 cm. Is this triangle a right triangle? If so, which side is the hypotenuse?

Solution:

We know that hypotenuse is the longest side. If 8 cm, 15 cm and 17 cm are the lengths of angled triangle, then 17 cm will be the hypotenuse.

Using the converse of Pythagoras theorem, we get

(17)² = (15)² + (8)²

⇒ 289 = 225 + 64

⇒ 289 = 289

Since, both the sides are equal therefore, 8 cm, 15 cm and 17 cm are the side of the right angled triangle having hypotenuse 17 cm.

3. The side of a triangle are of length 9 cm, 11 cm and 6 cm. Is this triangle a right triangle? If so, which side is the hypotenuse?

Solution:

We know that hypotenuse is the longest side. If 9 cm, 11 cm and 6 cm are the lengths of angled triangle, then 11 cm will be the hypotenuse.

Using the converse of Pythagoras theorem, we get

(11)² = (9)² + (6)²

⇒ 121 = 81 + 36

⇒ 121 ≠ 117

Since, both the sides are not equal therefore 9 cm, 11 cm and 6 cm are not the side of the right angled triangle.

The above examples of the converse of Pythagorean Theorem will help us to determine the right triangle when the sides of the triangles will be given in the questions.

Congruent Shapes

Congruent Line-segments

Congruent Angles

Congruent Triangles

Conditions for the Congruence of Triangles

Side Side Side Congruence

Side Angle Side Congruence

Angle Side Angle Congruence

Angle Angle Side Congruence

Right Angle Hypotenuse Side congruence

Pythagorean Theorem

Proof of Pythagorean Theorem