Two circles touch each other externally at A. Their two common tangents that do not pass through A meet at X and touch the larger circle at B and C. If the radius of the larger circle is 15 cm and CX = 20 cm, what is the radius of the smaller circle (in cm)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3.75
The two tangents meet on the line joining the centres, and a radius meets a tangent at 90°. The right triangles formed by X, each centre and its point of contact on one tangent are similar, so the distances from X are in the ratio of the radii.
Let the centres be O (larger) and O′ (smaller), smaller radius \(r\)
\(\angle OCX=90^\circ\): \(OX=\sqrt{15^{2}+20^{2}}=25\,\mathrm{cm}\)
The circles touch, so \(OO'=15+r\) and \(O'X=25-(15+r)=10-r\)
Similar triangles: \(\displaystyle \frac{O'X}{OX}=\frac{r}{15}\), so \(\displaystyle \frac{10-r}{25}=\frac{r}{15}\)
\(150-15r=25r\), so \(\displaystyle r=\frac{150}{40}=3.75\,\mathrm{cm}\)
