A triangle $ABC$ is drawn to circumscribe a circle of radius $4$ cm such that the segments $BD$ and $DC$ are of lengths $10$ cm and $8$ cm respectively. Find the lengths of the sides $AB$ and $AC$, if

Question

A triangle $ABC$ is drawn to circumscribe a circle of radius $4$ cm such that the segments $BD$ and $DC$ are of lengths $10$ cm and $8$ cm respectively. Find the lengths of the sides $AB$ and $AC$, if it is given that area $	riangle ABC = 90$ cm$^2$.

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Accepted Answer

Step 1: Triangle ABC with incircle, tangent points D, E, F and side lengths labeled with variable x
Step 2: Join $OA$, $OB$, $OC$ and draw $OE \perp AC$ and $OF \perp AB$.
$BF = 10$ cm, $CE = 8$ cm, Let $AF = AE = x$
Step 3: ar $	riangle ABC$ = ar $	riangle BOC$ + ar $	riangle COA$ + ar $	riangle AOB$
$90 = \frac{1}{2} 	imes 4 	imes (BC + CA + AB)$
$90 = 2(18 + 8 + x + 10 + x)$
Step 4: $90 = 4(18 + x) \Rightarrow x = 4.5$
$AB = 10 + 4.5 = 14.5$ cm and $AC = 8 + 4.5 = 12.5$ cm

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