UPSC CDS 1 2022 Math Paper

Given: The sides of the triangle are $4\,\text{cm}, 7.5\,\text{cm}$ , and $8.5\,\text{cm}$ Concept used: Inradius: - The inradius of a triangle is formed by first dividing each of the three angles in half. - The point at which these three lines meet is the center of the incircle. - The inradius is a line drawn from the center to perpendicularly intersect a side of the triangle. - In-radius of a right angle triangle $(r)=\frac{(P+B-H)}{2}$ Here, $P=$ Perpendicular, $B=\text{Base}, H=$ Hypotenuse Calculation: Since $4^{2}+7.5^{2}=8.5^{2}$ $\Rightarrow 16+56.25=72.25$ $\Rightarrow 72.25=72.25 \;\text{ (Satisfying) }\;$ Hence, the sides of the triangle are Pythagorean triplets According to the concept, In $\triangle ABC$ in-radius of the circle $\Rightarrow r=\frac{AB+BC-AC}{2}$ $\Rightarrow r=\frac{4+7.5-8.5}{2}$ $\Rightarrow r=\frac{3}{2}=1.5$ $\therefore$ The radius of the circle inscribed is $1.5\,\text{cm}$ .