Question

The hypotenuse of a right triangle is 4 cm more than its shorter side if the third Sideis 2 cm less than its hypotenuse find all sides of the triangle

Answer 1

Gɪᴠᴇɴ :-

The hypotenuse of a right triangle is 4 cm more than its shorter side if the third Sideis 2 cm less than its hypotenuse.

ᴛᴏ ғɪɴᴅ :-

• Length of all sides of triangle

sᴏʟᴜᴛɪᴏɴ :-

➦ Let shorter side be x cm

Then,

➦ Hypotenuse = (x + 4) cm

➦ Third side = ( x + 4) - 2 = ( x + 2) cm

On using Pythagoras theorem, we get,

➦ Hypotenuse² = Base² + Altitude²

➭ (x + 4)² = x² + (x +2)²

➭ x² + 16 + 8x = x² + x² + 4 + 4x

➭ x² + 16 - 4 = 2x² + 4x - 8x

➭ 12 = 2x² - x² - 4x

➭ x² - 4x - 12 = 0

➭ x² - 6x + 2x - 12 = 0

➭ x(x - 6) + 2(x - 6) = 0

➭ (x + 2)(x - 6) = 0

➭ x = -2 or x = 6

Hence,

Length can't be negative.

So, we take x = 6

Now,

• Shorter side = x = 6cm

• Hypotenuse = (x + 4) = 6 + 4 = 10cm

• Third side = (x + 2) = 6 + 2 = 8cm

Answer 2

Given :-

• the hypotenuse of a right triangle is 4 cm more than its shorter side if the third side is 2 cm less than its hypotenuse.

To find :-

• Length of all sides of triangle.

Solution :-

→ Let shorter side be x cm.

Then,

→ Hypotenuse = (x + 4) cm.

→ Third side = (x + 4) - 2 = (x + 2) cm.

On using Pythagoras theorem, we get,

→ Hypotenuse² = Base² + Altitude²

➜ (x + 4)² = x² + (x + 2)²

➜ x² + 16 + 8x = x² + x² + 4 + 4x

➜ x² + 16 - 4 = 2x² + 4x - 8x

➜ 12 = 2x² - x² - 4x

➜ x² - 4x - 12 = 0

➜ x² - 6x + 2x - 12 = 0

➜ x(x - 6) + 2(x - 6) = 0

➜ (x + 2)(x - 6) = 0

➜ x = -2 or x = 6

Hence,

• Shorter side = x = 6cm.

• Hypotenuse = (x + 4) = 6 + 4 = 10cm.

• Third side = (x + 2) = (6 + 2) = 8cm.