Question

A triangle of area 84 cm² has a perimeter of 56 cm. If the largest side measures 25 cm, then find the sum (in cm) of the largest and the smallest side of the triangle 
1. 25 cm
2. 32 cm
3. 30 cm
4. 20 cm

Answer 1

Correct Answer - Option 2 : 32 cm

Given:

The perimeter of a triangle is 56 cm and its area is 84 cm2.

Formula used:

Semi perimeter of a Δ = (a + b + c)/2

Area of Δ = √{s(s - a)(s - b)(s - c)}

Where a, b, c are sides of the Δ.

s → semiperimeter

Calculations:

Let the smallest side of the triangle be x cm

then the second side of the triangle = 56 - 25 - x = (31 - x) cm

Semi perimeter of the Δ = 56/2 = 28 cm

so area of the Δ = √{s(s - a)(s - b)(s - c)}

But area of the Δ is given in the question.

Thus √{s(s - a)(s - b)(s - c)} = 84

⇒ √[28(28 - 25){28 - (31 - x)}(28 - x)] = 84

Squaring both sides, we'll get

⇒ 84 × (28 – x) × (x – 3) = 84² 

⇒ (28 – x) × (x – 3) = 84

⇒ x² – 31x + 168 = 0

⇒ x² – 24x – 7x + 168 = 0

⇒ x(x – 24) – 7(x – 24) = 0

⇒ (x – 7)(x – 24) = 0

⇒ x = 7 or 24

If x = 24, then 31 - 24 = 7. This contradicts our assumption.

And if x = 7, then 31 - 7 = 24. This satisfies our assumption.

Thus the required sum = 25 + 7 = 32 cm

∴ The sum of the largest and the smallest side of the triangle is 32 cm.