In an obtuse-angled triangle ABC, the length of its longest side AB is 50 cm and one of the other two sides is 42 cm. If the area of the triangle is 294 cm2 , what is the length (in cm) of its third side?

Question

Question

This question was previously asked in

SSC CHSL Exam 2024 Tier-I Official Paper (Held On: 11 Jul, 2024 Shift 4)

Answer 1

Given:

In an obtuse-angled triangle ABC,

Length of the longest side AB = 50 cm

One of the other two sides = 42 cm

Area of the triangle = 294 cm²

Formula used:

Area of a triangle = (1/2) × base × height

Using Heron's formula for the area of a triangle:

Area = √(s(s-a)(s-b)(s-c))

where s = (a+b+c)/2 is the semi-perimeter.

Calculation:

Let the third side be c cm.

Given area = 294 cm²

Using the area formula:

294 = (1/2) × 50 × h

⇒ h = (294 × 2)/50 = 588/50 = 11.76 cm

Now, using Heron's formula:

Semi-perimeter s = (50 + 42 + c)/2

Area = √((92+c)/2 ((92+c)/2 - 50) ((92+c)/2 - 42) ((92+c)/2 - c)) = 294

Let's simplify this step-by-step:

294 = √((92+c)/2 ((92+c)/2 - 50) ((92+c)/2 - 42) ((92+c)/2 - c))

Squaring both sides:

294² = (92+c)/2 ((92+c)/2 - 50) ((92+c)/2 - 42) ((92+c)/2 - c)

Simplify to find c.

On solving, we find:

The length of the third side is 2√(58) cm.