In the figure, 𝑃𝑄 and 𝑃𝑅 are
two tangents to a circle with centre 𝑂. If the measure of the angle 𝑄𝑃𝑅
is equal to 46 degrees, find the measure of the angle 𝑄𝑂𝑅.
Let’s begin by listing what we know
about this diagram. First, since 𝑂 is the centre of
the circle and 𝑄 and 𝑅 are two points on its circumference, we know that the lines
𝑂𝑄 and 𝑂𝑅 are the radii of the circle. This means that 𝑂𝑄 and 𝑂𝑅 are
of equal lengths. We also know that they meet the
tangents at 90 degrees, since a radius and a tangent are always perpendicular.
Notice, we now have a
quadrilateral. Angles in a quadrilateral add to
360 degrees. So we can calculate the measure of
angle 𝑄𝑂𝑅 by subtracting 90 and 90 and 46 from 360. 90 plus 90 plus 46 is equal to
226. And 360 minus 226 is 134. The measure of the angle 𝑄𝑂𝑅 is