Multiple choice

Two chords $AB$ and $CD$ of circle whose centre is $O$, meet at the point $P$ and $\angle AOC=50^{o}, \angle BOD=40^{o}$. Then the value of $\angle BPD$ is:

  1. $60^{o}$
  2. $40^{o}$
  3. $45^{o}$
  4. $75^{o}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The angle subtended by an arc at the center is double the angle subtended at the circumference. Angle ACB = 1/2 * angle AOB = 1/2 * (50 + 40) = 45 degrees. In triangle PBC, the exterior angle BPD = angle PCB + angle PBC, but it is simpler to note that angle BPD = 1/2 * (angle AOC + angle BOD) = 1/2 * (50 + 40) = 45 degrees.

AI explanation

For two chords AB and CD intersecting externally at point P, the angle between the secants is half the positive difference of the intercepted arcs. The measure of arc CA is equal to the central angle AOC, which is 50 degrees, and the measure of arc BD is equal to the central angle BOD, which is 40 degrees. Placing these values into the formula gives angle BPD as half of (50 degrees minus 40 degrees). Therefore, the value of angle BPD is 45 degrees.

Keep practicing — related questions