Question

$△ABC$ is isosceles with AB = AC. If a circle through B touches the side AC at its mid-point D and intersects the side AB at P, then find the value of $\frac{AP}{AB}$.

Answer 1

AB = AC (given, $\Delta ABC$ is isosceles) ...(i)

D is midpoint of AC (given)

$AD=\frac{AC}{2}=\frac{AB}{2}$ ……. From (i)

AD is tangent to the circle (given)

$\therefore A{D}^2=AP\times AB$

${\left(\frac{AB}{2}\right)}^2=AP\times AB$

$\frac{A{B}^2}{4}=AP\times AB$

$\frac{AP}{AB}=\frac{1}{4}$