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August 28, 2018

Area and inradius - Pre RMO 2018 Problem 2 Discussion


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 close to $r$?

PRMO 2018 Problem 2

Hint 1:

First, notice that the quadrilateral is a kite. Diagonals of a kite bisect each other (Prove this!) If $X$ is the point of intersection then it is easy to show $\Delta AXB \equiv \Delta AXD \implying BX = XD = 12$.

Finally to find the incenter, draw the angle bisector of $ \angle ADC$. Wherever it meets $AC$, call that point $I$. It is the incenter (Prove this too!) To draw the incircle, drop a perpendicular from $I$ to $AD$ and draw a circle taking that perpendicular segment as radius, and $I$ as the center.

Hint 2 -

Find out $AC$ It is easy to find (using Pythagoras theorem) that $AX = 5$ and $CX = 16$.

(Notice that $\Delta AXB$ is right angled. Hence $AX^2 + BX^2 = AB^2$ or $AX^2 = 13^2 - 12^2 = 5^2$ )

Hint 3 -

inradius and area is related Any polygon with an inscribable circle (a circle that touches each of its sides) has this beautiful property:  Area = $r \times s$ where $r$ = radius of the inscribed circle and $s$ = semiperimeter (half of the perimeter).

Why is that true? Well, there is a simple argument that works for all polygons (in which circles can be inscribed).

However we will describe the special case of this kite:

PRMO problem 2 2

Drop perpendiculars from incenter $I$ to each of the sides of the kite. Since the incircle touches each side, hence each side is a tangent to the incircle. Hence these perpendicular segments are clearly the radii of the incircle (after this was our method of construction) in the first place. Join $ID$ and $IB$.

The kite is thus divided into four triangles: red, green, yellow and blue. Notice that area of these triangles are  $\frac{1}{2} \times r \times AB, \frac{1}{2} \times r \times BC, \frac{1}{2} \times r \times CD, \frac{1}{2} \times r \times DA$ respectively.

Adding the areas of the $4$ triangles we will get the area of the kite which is: $\frac{1}{2} \times r \times (AB + BC + CD + DA ) =  r \times s$

Thus the area of the kite is in radius times semiperimeter. The exact same argument holds for any polygon with inscribable circles.

The semiperimeter is $\frac{1}{2} \times (13 + 13 + 20 + 20) = 33$. Hence the area of the kite is $33r$.

But we also know that area of the kite is the product of diagonals by $2$ (or just separately find out the area of the triangles $ABC, ADC$ and add them).

This gives the area to be  $\frac{1}{2} \times 24 \times 21 = 252$

Equating we have $33r = 252 or r ~ 7.6$. Its closest integer is $8$

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