Question

A circle has two equal chords $PQ$ and $PR$, diameter $PD$ cuts $QR$ in $E$. If $PR=12cm$ and $PE=8cm$, then the length of $PD$is ?

• A. $25$ cm
• B. $22$ cm
• C. $20$ cm
• D. $18$ cm

Answer 1

The correct option is D $18$ cm

Given that PD is diameter and it bisects the chord QR

$⟹PD\perp QR$ because any line bisecting a chord and passing through center is perpendicular to the chord.

In $△PER$

$\angle PER=90$

$⟹R{E}^2=P{R}^2–P{E}^2={12}^2–8^2$

$RE=\sqrt{80}=EQ$

When two chords intersect, the product of the segments of one chord is equal to the product of segments of the other.

$⟹PE\times ED=RE\times EQ$

$ED=\frac{\sqrt{{80}^2}}{8}=10$

$PD=PE+ED=18cm$