The sides of a triangle are 109 cm, 171 cm, and 100 cm. What is the length (in cm) of its altitude corresponding to the side with a length of 171 cm?​

Correct option is C

Given:

The sides of the triangle are 109 cm, 171 cm, and 100 cm.

We need to find the length of the altitude corresponding to the side of length 171 cm.

Formula Used:

Area of triangle = $$\sqrt{s(s - a)(s - b)(s - c)}$$​

Where s is the semi-perimeter of the triangle:

s = $$\frac{a + b + c}{2}$$​

Area of a triangle in terms of the base and height:

A = $$\frac{1}{2} \times \text{Base} \times \text{Height}$$​

Solution:

a = 109 cm, b = 171 cm, c = 100 cm

$$s = \frac{109 + 171 + 100}{2} = \frac{380}{2} = 190 \, \text{cm}$$

​$$\\ \ \\A = \sqrt{190(190 - 109)(190 - 171)(190 - 100)}\\ \ \\A = \sqrt{190 \times 81 \times 19 \times 90}\\ \ \\ A = 5130 \, \text{cm}^2$$​​

​Now, 

​$$A = \frac{1}{2} \times 171 \times h\\ \ \\5130 = \frac{1}{2} \times 171 \times h\\ \ \\5130 = 85.5 \times h$$​​

h = $$\frac{5130}{85.5}= 60 \, \text{cm}$$