Question

The area of a triangle is 96 cm2 and the ratio of its sides is 6 ∶ 8 ∶ 10. What is the perimeter of the triangle?

• A. 48 cm
• B. 56 cm
• C. 64 cm
• D. 44 cm

Answer 1

Answer:

A. 48 cm

Solution:

Given:

The area of a triangle is 96 cm2 and the ratio of its sides is 6 ∶ 8 ∶ 10.

Concept used:

Area of a triangle $=\sqrt {S (S - A) (S - B) (S - C)}$ (S = Semi-perimeter = (A + B + C)/2, where A, B, C are the measure of the sides of the triangle)

Calculation:

Let the sides of the triangle be 6k, 8k, and 10k respectively.

Perimeter of the triangle = (6k + 8k + 10k) = 24k

Semi-perimeter $= \frac{24k}{2} = 12k$

​According to the concept,

$\sqrt {12k (12k - 6k) (12k - 8k) (12k - 10k) } = 96$

$\sqrt {12k \times 6k \times 4k \times 2k} = 96$

$\sqrt {144 \times 4 \times k^4} = 96$

⇒ 24k² = 96

⇒ k² = 96/24

⇒ k² = 4

⇒ k = 2

⇒ 24k = 48

∴ The perimeter of the triangle is 48 cm.

 Alternate Method

The ratio of sides of the triangle is 6 ∶ 8 ∶ 10, [ as we know 6, 8, 10 are Pythagorean triplets]

Let the sides of the triangle be 6k, 8k, and 10k respectively.

So, the triangle is a right-angle triangle, area = 1/2 × base × height  

⇒ 1/2 × 6k × 8k = 96 

⇒ 2k = 4 

⇒ k = 2

So, the perimeter of the triangle is ⇒ (6k + 8k + 10k) = 24k = 24 × 2 = 48 cm