Multiple choice

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

  1. $25$ cm
  2. $22$ cm
  3. $20$ cm
  4. $18$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let O be the center. PQ=PR, so O lies on the angle bisector of QPR. PD is diameter, so PD passes through O. Triangle PQR is isosceles. E is on QR. Since PD is diameter and perpendicular to chord QR (as it bisects it), PE is the altitude. In right triangle PQR, if we use properties of chords, the length of PD can be determined by geometric relations.

AI explanation

Since equal chords PQ and PR subtend equal angles at the center, the diameter PD bisects the chord QR at E. By the Intersecting Chords Theorem, PE multiplied by ED equals QE squared, so 8 times ED equals 6 squared which is 36, giving ED as 4.5 cm. The total diameter PD is PE plus ED, which is 8 plus 4.5, making the length 18 cm.

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