If O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50 degree with PQ, then find angle POQ.
100
To find the measure of angle POQ, we need to first understand the relationships between angles in a circle.
1. An angle formed by a tangent and a chord at the point of tangency is equal to half the measure of the intercepted arc (angle = 1/2 * intercepted arc).
2. The measure of an angle at the center of a circle is equal to twice the measure of the intercepted arc (angle = 2 * intercepted arc).
Now let's apply these properties to the given problem:
1. The tangent PR at point P makes an angle of 50 degrees with PQ. According to property 1, this angle is equal to half the measure of the intercepted arc QR.
2. Since angle PRQ is a straight angle (180 degrees), we can find the measure of arc QR by subtracting the angle measure from 180: Arc QR = 180 - 50 = 130 degrees.
3. Now, according to property 2, the measure of angle POQ is twice the measure of arc QR. Thus, angle POQ = 2 * 130 = 260 degrees.
Therefore, angle POQ measures 260 degrees.