In the given figure PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°. Find ∠PCA.

Given that PQ is a tangent at point C to a circle with center O, and AB is a diameter with ∠CAB = 30°, we need to find ∠PCA.

To find ∠PCA, we can use the properties of tangents and angles in a circle.

Step 1: Draw a diagram representing the given information. Label the points as described: PQ is the tangent at point C, AB is the diameter, and O is the center of the circle.

Step 2: Since AB is a diameter, it passes through the center O. Therefore, ∠CAB is an inscribed angle that intercepts the same arc as ∠COB.

Step 3: By the inscribed angle theorem, we know that the measure of an inscribed angle is equal to half the measure of its intercepted arc. Therefore, ∠CAB = 30° implies that the intercepted arc COB has a measure of 60°.

Step 4: Since PQ is a tangent to the circle at point C, it is perpendicular to the radius OC. Therefore, ∠PCO is a right angle.

Step 5: Since ∠PCO is a right angle and ∠COB is 60°, we can find ∠PCA by subtracting the measures of ∠PCO and ∠COB from 180° (the sum of angles in a triangle).

Step 6: ∠PCA = 180° - ∠PCO - ∠COB = 180° - 90° - 60° = 30°.

Therefore, ∠PCA is equal to 30°.

Sure, let's solve the problem step by step.

Step 1: Draw the given figure with a circle, tangent PQ, diameter AB, and angle CAB.

Step 2: Since AB is a diameter, it passes through the center of the circle, which we'll call O.

Step 3: Since PQ is a tangent to the circle at point C, we know that angle ACB is 90 degrees.

Step 4: Since AB is a diameter, angle ACB is subtended by the diameter AB, which means it is a right angle.

Step 5: Since angle CAB is given as 30 degrees, we can use the fact that the sum of angles in a triangle is 180 degrees to find angle BAC.

Step 6: Angle BAC + angle CAB + angle ABC = 180 degrees.

Step 7: Substituting the given values, we have angle BAC + 30 degrees + 90 degrees = 180 degrees.

Step 8: Simplifying the equation, we have angle BAC + 120 degrees = 180 degrees.

Step 9: Subtracting 120 degrees from both sides, we have angle BAC = 60 degrees.

Step 10: Since angle BAC is an exterior angle of triangle APC, we can use the fact that the exterior angle is equal to the sum of the opposite interior angles to find angle PCA.

Step 11: Angle PCA + angle BAC = angle APC.

Step 12: Substituting the known values, we have angle PCA + 60 degrees = 180 degrees.

Step 13: Subtracting 60 degrees from both sides, we have angle PCA = 120 degrees.

Therefore, the measure of angle PCA is 120 degrees.