Question

Find the radius of the incircle of the right angled triangle whose sides measure \[8,10,12\] \[cm\]?
\[\begin{align}
  & A.1.22cm \\
 & B.2.66cm \\
 & C.3.41cm \\
 & D.4.51cm \\
\end{align}\]

Answer 1

Hint: Since we are given with the sides of the right angles triangle, we can find the radius of the incircle by substituting the values in the formula. Firstly, we have to find the area of the triangle and then the sum of the sides of the triangle. These obtained values must be substituted in the incircle formula and we will be obtaining the required value.

Complete step-by-step solution:
Now let us have a brief regarding the incircle. An incircle can also be termed as an inscribed circle. It is generally the largest circle contained in a triangle. This circle touches all the three sides of the triangle. The centre of the circle is the same as the centre of the triangle and it is called the triangl’s incentre. In general, an incircle is formed by the intersection of all the three angular bisectors of the triangle.
Now let us find the radius of the incircle of the right angled triangle whose sides measure \[8,10,12\] \[cm\].
The formula of the radius of the incircle when sides are given is \[\dfrac{2\times \text{area of triangle}}{\text{sum of sides}}\]
Firstly, let us find the area of the triangle.
Area of the triangle \[=\dfrac{1}{2}bh=\dfrac{1}{2}\times 8\times 10=40c{{m}^{2}}\]
Now let us find the sum of the sides of the triangle. We get,
Sum of the sides= \[8+10+12=30cm\]
Now let us substitute these obtained values in the formula of radius of the incircle
\[\Rightarrow \dfrac{2\times \text{area of triangle}}{\text{sum of sides}}=\dfrac{2\times 40}{30}=\dfrac{8}{3}=2.66cm\]
\[\therefore \] The radius of the incircle is \[2.66cm\]
Hence option B is the correct answer.

Note: While calculating the area of the triangle, we must correctly consider the values of the base and height because incorrect values considered will retain us with incorrect answers. We must always have a note that the hypotenuse is the longest side of a right angled triangle, so we must assign the larger value to the hypotenuse.