Question

What is the inradius of a right triangle with a height of $3$, base of $4$ and a hypotenuse of $5$?
$1)1$
$2)2$
$3)3$
$4)1.5$

Answer 1

Hint: First, we need to know about the concept of the area of the triangle.
The total surface or the space enclosed by the three boundaries of the triangle is known as the area of the triangle. If the base and height of the triangle is given then we can able to find the area of the triangle using the formula that $\dfrac{1}{2}bh$
Formula used:
Radius $r = \dfrac{A}{S}$ , where A is the area of the triangle and S, is the semi perimeter.

Complete step-by-step solution:
Since from the question given that we have a right triangle with a height of $3$, the base of $4$ , and a hypotenuse of $5$
Thus, we need to find the inradius of the right triangle.
We know that the area of the triangle formula is $\dfrac{1}{2}bh$ and height is $3$, the base is $4$
Thus, substituting the values, we get $\dfrac{1}{2}bh = \dfrac{1}{2}(3 \times 4)$ and further solving we get $\dfrac{1}{2}bh = 6$
Also, we know that the semi perimeter is the half of the sum of all sides of the given triangle represented as $\dfrac{1}{2}(a + b + c)$
Hence substituting the values, we have $\dfrac{1}{2}(a + b + c) = \dfrac{1}{2}(3 + 4 + 5)$
Further solving we get $\dfrac{1}{2}(a + b + c) = 6$
Hence, we get the Area of the triangle as $6$ and then the semi perimeter as $6$
Hence substituting the values into the radius we have $r = \dfrac{A}{S} = \dfrac{6}{6} = 1$
Thus, the option $1)1$ is correct.

Note: We also need to know about the concept of the semi perimeter.
Perimeter is the sum of all sides of the given triangle and the semi perimeter is half of the sum of all sides of the given triangle. And hence which can be represented as $\dfrac{1}{2}(a + b + c)$ where “a” is the height, “b” is the base and “c” is the hypotenuse (assumption and it can represent any side).