The sides of a triangle are 15 cm,28 cm, and 41 cm. What is the length of its altitude corresponding to the side with a length of 28 cm ?

Correct option is D

Given:​​

The sides of the triangle are a = 15 cm, b = 28 cm, and c = 41 cm.

length of the altitude corresponding to the side of length 28 cm = ?

Concept Used:

Heron's Formula 

Area  $$A = \sqrt{s(s - a)(s - b)(s - c)}$$ 

Where:

​$a,b,c$ are the lengths of the sides of the triangle,

​$s$ is the semi-perimeter of the triangle, calculated as: $$s = \frac{a + b + c}{2}$$

Area of triangle $$= \frac{1}{2} \times \text{base} \times \text{height}$$ 

Solution: 

The semi-perimeter 's' of the triangle is: 

​$$s = \frac{15 + 28 + 41}{2} = 42 \, \text{cm}$$ 

​Now, we calculate the area $A$ using Heron's formula: 

​$$A = \sqrt{42(42 - 15)(42 - 28)(42 - 41)}$$ 

$$A = \sqrt{42 \times 27 \times 14 \times 1}$$ 

$$A = \sqrt{15876} = 126 \,cm^2$$​​

The area of a triangle can also be calculated using the formula: 

​$$A= \frac{1}{2} \times \text{base} \times \text{height}$$ 

Let the altitude corresponding to the base of 28 cm be $h$. Then, we have:

$$126 = \frac{1}{2} \times 28 \times h$$ 

$$126 = 14h$$ 

$$h = \frac{126}{14} = 9 \, \text{cm}$$ 

​Thus the length of the altitude corresponding to the side of length 28 cm is 9 cm.​