Hmm... Tangents
Geometry
Level
2
Let ABC be a triangle with AB=8 cm, BC=12 cm and AC=9 cm. Let PQR be the incircle of triangle ABC such that P, Q and R are points tangent to lines AB, BC and AC respectively. Find the length of AP.
The answer is 2.5.
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1 solution
Let the length of AP be x cm. Then, PB= (8-x) cm.Since the lengths of any two lines connecting a tangent of a circle to a point outside the line are equal, therefore AP=AR, BP=BQ and CQ=CR. Hence, we have PB=BQ=(8-x) cm. Subsequently, CQ=BC-BQ=(4+x) cm. We know that CQ=CR=(4+a) cm.Then, AR=AC-CR=(5-x) cm. Since AR=AP, we know that x cm=(5-x) cm. From the equation, we get x= 2 . 5 c m .