Question

In the given figure, PQ is a transverse common tangent to two circles with centers A and B and of radii 5 cm and 3 cm respectively. If PQ intersects AB at C such that CP = 12 cm, calculate AB.

Answer 1

In $△$CPA, applying Pythagoras theorem

${CA}^2$ = ${AP}^2$ + ${CP}^2$

${CA}^2$ = $5^2$ + ${12}^2$

${CA}^2$ = 169

CA = 13

We know that

$\angle$CQB = $\angle$CPA = ${90}^{\circ }$

$\angle$QCB = $\angle$PCA (Vertically opposite angle)

Therefore $△$CPA ~ $△$CQB

$\Rightarrow$ $\frac{QB}{PA}$ = $\frac{CB}{CA}$

CB = $\frac{(QB\times CA)}{PA}$

CB = $\frac{(3\times 13)}{5}$

CB = 7.8

AB = AC + BC = 13 + 7.8 = 20.8 cm