Question

What is the radius of a circle inscribed in a triangle having side lengths $35\mathrm{cm},44\mathrm{cm}$and $75\mathrm{cm}$?

• A. $3cm$
• B. $4cm$
• C. $5cm$
• D. $6cm$

Answer 1

The correct option is D

$6cm$

Find the radius of a circle.

Explanation for the correct option:

Option (D).

The triangle's given side lengths are $35\mathrm{cm},44\mathrm{cm}$ and $75\mathrm{cm}$.

The Semiperimeter of the triangle is calculated as follows:

$\begin{array}{rcl}\mathrm{s}& =& \frac{(35+44+75)}{2}\\ & =& 154/2\\ & =& 77\mathrm{cm}\end{array}$

Area of a triangle:

$\begin{array}{rcl}\mathrm{A}& =& \sqrt{\left(\mathrm{s}\right(\mathrm{s}-\mathrm{a})(\mathrm{s}-\mathrm{b})(\mathrm{s}-\mathrm{c}\left)\right)}\\ & =& \sqrt{(77\times 42\times 33\times 2)}\\ & =& 462{\mathrm{cm}}^2\end{array}$

$\begin{array}{rcl}\mathrm{Radius}\mathrm{of}\mathrm{incircle}& =& \frac{\mathrm{area}}{\mathrm{semi}\mathrm{perimeter}}\\ & =& \frac{462}{77}\\ & =& 6\mathrm{cm}\end{array}$

Thus, the radius of a circle is $6\mathrm{cm}$.

Hence, Option (D) is the correct answer.

Explanation of the incorrect option:

Since there can be only one circle inscribed in the given triangle such that it touches its three sides whose radius is $6\mathrm{cm}$ as determined above.

So, Option (A), Option (B) and Option (C) are incorrect options.

Hence, Option (D) is the correct answer.