Multiple choice

A right angled triangle ABC right angled at A and $b = 18cm$ and $c = 63cm$. The circle $C_1$ is drawn with centre lying on hypotenuse BC and touching the side AB and CA. Another circle $C_2$ is drawn concentric with $C_1$ and passing through A. The area of the circle $C_2$ is

  1. $144 \pi cm^2$
  2. $196 \pi cm^2$
  3. $288 \pi cm^2$
  4. $392 \pi cm^2$
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D Correct answer
AI explanation

Let the center of C1 on the hypotenuse BC be point O, where it touches AB at D and CA at E, making OD = OE = r. In right triangle ODB, we have angle OBD equal to angle ABC, so sin(angle ABC) = r/(c-r); similarly in right triangle OEC, cos(angle ABC) = r/(b-r). This yields r = bc/(b+c), and substituting b = 18 and c = 63 gives r = 1134/81 = 14. Since the length of the hypotenuse BC is sqrt(18^2 + 63^2) = 45*sqrt(5) and sin(angle ABC) is 18/sqrt(4185), the distance OA is found using OA = (c-r)*cos(angle ABC) + (b-r)*sin(angle ABC). Calculating OA^2 gives the area of circle C2 as pi*OA^2 = 392*pi.

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