A chord PQ is produced to R so that QR = r (radius of the circle). Through R, the diameter AB is drawn cutting the circle in A and B such that arc BP = x.arc AQ then find the value of x.
Question
A chord PQ is produced to R so that QR = r (radius of the circle). Through R, the diameter AB is drawn cutting the circle in A and B such that arc BP = x.arc AQ then find the value of x.
Solution
The problem is a geometry problem related to circles. Here are the steps to solve it:
Step 1: Draw a diagram based on the problem description. Draw a circle with center O, a chord PQ extended to R such that QR equals the radius of the circle. Draw the diameter AB through R, cutting the circle at points A and B.
Step 2: Since QR is the radius of the circle, triangle ORQ is an isosceles triangle (because OR = OQ = r). Therefore, ∠ORQ = ∠OQR.
Step 3: Also, ∠OQR = ∠OAB (because they are corresponding angles). Therefore, ∠ORQ = ∠OAB.
Step 4: Since ∠ORQ = ∠OAB, arc AQ = arc BP (because the measure of an arc is equal to the measure of its central angle). Therefore, x = 1.
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