Exam

Olympiad workbook

Subject

IOQM

Chapter

Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

In a quadrilateral A B C D , it is given that A B=A D=13, B C=C...

In a quadrilateral $A B C D$, it is given that $A B=A D=13, B C=C D=20, B D=24$. If $r$ is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to $r$ ?

Step-by-step solution

$$\begin{aligned} \text { Area of } A B C D & =\text { Area of } \triangle A B D+\text { Area of } \triangle B C D \\ & =\sqrt{25 \times 12 \times 12 \times 1}+\sqrt{32 \times 8 \times 12 \times 12} \\ & =60+192=252 \\ \text { Inradius(r) } & =\frac{\text { Area }}{\text { semi-perimeter }}=\frac{252}{33}=\frac{84}{11}=7 . \overline{64} \end{aligned}$$ Hence integer nearest to $r$ is 8.