**Identifying a right triangle :**

We can use the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle when the lengths of the three sides are given.

The Pythagorean Theorem states that if a triangle is a right triangle, then, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

That is, if a and b are legs and c is the hypotenuse, then

a² + b² = c²

The converse of the Pythagorean Theorem states that if a² + b² = c², then the triangle is a right triangle.

**Example 1 :**

Determine whether triangle with the side lengths given below is a right triangle.

9 inches, 40 inches, and 41 inches

**Solution : **

**Step 1 :**

Let a = 9, b = 40, and c = 41.

(Always assume the longest side as "c")

**Step 2 :**

Find the value of a² + b².

a² + b² = 9² + 40²

a² + b² = 81 + 1600

a² + b² = 1681 ----- (1)

**Step 3 :**

Find the value of c².

c² = 41²

c² = 1681 ----- (2)

**Step 4 :**

From (1) and (2), we get

a² + b² = c²

By the converse of Pythagorean theorem, the triangle with the side lengths 9 inches, 40 inches, and 41 inches is a right triangle.

**Example 2 :**

Determine whether triangle with the side lengths given below is a right triangle.

8 meters, 10 meters, and 12 meters

**Solution : **

**Step 1 :**

Let a = 8, b = 10, and c = 12.

(Always assume the longest side as "c")

**Step 2 :**

Find the value of a² + b².

a² + b² = 8² + 10²

a² + b² = 64 + 100

a² + b² = 164 ----- (1)

**Step 3 :**

Find the value of c².

c² = 12²

c² = 144 ----- (2)

**Step 4 :**

From (1) and (2), we get

a² + b² ≠ c²

By the converse of Pythagorean theorem, the triangle with the side lengths 8 meters, 10 meters, and 12 meters is not a right triangle.

**Example 3 :**

Determine whether triangle with the side lengths given below is a right triangle.

14 cm, 23 cm, and 25 cm

**Solution : **

**Step 1 :**

Let a = 14, b = 23, and c = 25.

(Always assume the longest side as "c")

**Step 2 :**

Find the value of a² + b².

a² + b² = 14² + 23²

a² + b² = 196 + 529

a² + b² = 725 ----- (1)

**Step 3 :**

Find the value of c².

c² = 25²

c² = 625 ----- (2)

**Step 4 :**

From (1) and (2), we get

a² + b² ≠ c²

By the converse of Pythagorean theorem, the triangle with the side lengths 14 cm, 23 cm, and 25 cm is not a right triangle.

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