Pre RMO 2018(IOQM), Question 2 incircle quadrilateral

Question 2, Pre RMO 2018-19(IOQM):

In a quadrilateral ABCD, it is given that AB = AD = 13, BC= CD = 20, BD = 24. If r is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to r?.

Prerequisites:
1. Incenter, incircle and angle bisectors in triangle/polygons
2. Kite diagonals

Solution:

ABCD is a kite.

So, $AC \perp BD$.

So, $BO = OD = 12 \text{ cm}$.

$$AO = \sqrt{13^2 - 12^2} = 5 \text{ cm}$$ $$OC = \sqrt{20^2 - 12^2} = 16 \text{ cm}$$ $$\text{ar}(\square ABCD) = \tfrac{1}{2} \times AC \times BD$$ $$= \tfrac{1}{2} \times 21 \times 24$$ $$= 252 \text{ cm}^2$$ $$r = \frac{\Delta}{S} = \frac{252}{33} = 7.63 \text{ cm}$$

Closest integer $= 8$.