Video: CBSE Class X • Pack 3 • 2016 • Question 1

CBSE Class X • Pack 3 • 2016 • Question 1

02:22

Video Transcript

In a figure, 𝑃𝑄 is a tangent at point 𝐶 to a circle with centre 𝑂. If 𝐴𝐵 is a diameter and angle 𝐶𝐴𝐵 equals 30 degrees, find angle 𝑃𝐶𝐴.

Let’s label the information we know. We know that 𝑃𝑄 is a tangent at point 𝐶. And that means the measure of 𝑃𝐶𝑂 would be 90 degrees, because a tangent is perpendicular to a radius. We know that 𝐴𝐵 is a diameter. And that means 𝐴𝑂 and 𝑂𝐵 will be equal. They are both a radius of this circle. We also know that 𝑂𝐶 would be equal to the other two radii. We were also told that angle 𝐶𝐴𝐵 equals 30 degrees.

We’re searching for the measure of angle 𝑃𝐶𝐴. That’s here. What can we say about the measure of angle 𝑃𝐶𝐴? If we add angle 𝐴𝐶𝑂 to angle 𝑃𝐶𝐴, it will equal 90 degrees. We’ve already said that angle 𝑃𝐶𝑂 must measure 90 degrees, because the tangent line 𝑃𝐶 is perpendicular to the radius 𝑂𝐶. To find 𝑃𝐶𝐴, we need to consider if we know what angle 𝐴𝐶𝑂 would be.

Look at triangle 𝐴𝐶𝑂. Line segment 𝐴𝑂 is equal in length to line segment 𝑂𝐶 because they are both a radius of the circle. Because we know this, the measure of angle 𝐴𝐶𝑂 must be equal to 30 degrees. The opposite side length from the 30 degrees is equal in both cases.

We can take this information and plug it in to our original equation we wrote. Angle 𝑃𝐶𝐴 plus 30 degrees must equal 90 degrees. If we subtract 30 degrees from both sides, we find that the measure of angle 𝑃𝐶𝐴 is 60 degrees.

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