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.