In the following figure, two
tangents 𝑅𝑄 and 𝑅𝑃 are drawn from an external point 𝑅 to a circle with centre
𝑂. If the angle 𝑃𝑅𝑄 is equal to 120
degrees, prove that 𝑂𝑅 is equal to 𝑃𝑅 plus 𝑅𝑄.
Let’s begin by listing what we
already know about the various elements in this diagram. We can begin by adding the lines
that join 𝑂 and 𝑃 and 𝑂 and 𝑄. Remember since these are lines
joining the centre of the circle with a point on the circumference, they are both
the radii of the circle. This means they’re of equal
We can also use circle theorems to
help us identify some other key facts. We know that the radius and the
tangent meet at 90 degrees. So the measure of angle 𝑂𝑃𝑅 and
𝑂𝑄𝑅 equals 90 degrees. We also know that two tangents that
meet at the same point are of equal length. So the length of the line segment
𝑃𝑅 is equal to the length of the line segment 𝑄𝑅.
Notice now that we have two
right-angled triangles with a shared line 𝑂𝑅. Since these triangles share the
line 𝑂𝑅, which is their hypotenuse, and they have two more sides of equal length
as we’ve shown, they must be congruent triangles. Congruent triangles are
And since angle 𝑃𝑅𝑄 is 120
degrees, we can split this exactly in half to show that the angle 𝑃𝑅𝑂 is equal to
the angle 𝑄𝑅𝑂, which is equal to 60 degrees. Angles in a triangle add to 180
degrees. And since 180 minus 90 minus 60 is
30, this means that angle 𝑃𝑂𝑅 and 𝑄𝑂𝑅 is equal to 30 degrees.
Now, let’s redraw one of these
triangles. We can call the length 𝑃𝑅 𝑥 and
the length 𝑂𝑅 𝑦. These are two of the sides we’re
trying to find a relationship between. Since we have a right-angled
triangle for which we know the measure of the other angles, we can use right angle
trigonometry to form this relationship.
Let’s label this triangle from the
point of view of angle 𝑃𝑂𝑅. The line 𝑂𝑅 is the hypotenuse of
the triangle: that’s the one opposite the right angle. The side opposite the angle we’re
interested in is the opposite and the other side is the adjacent.
Since we’re looking to find a
relationship between the opposite and the hypotenuse in this triangle, we’ll use the
sin ratio. sin is equal to opposite over hypotenuse. Substituting what we know into this
formula gives us sin of 30 is equal to 𝑥 over 𝑦. We should know though that sin 30
is equal to a half. So we’re going to solve this
equation and make 𝑦 the subject.
To do this, we’ll first multiply
both sides by 𝑦 to give us 𝑦 over two is equal to 𝑥. And then, we’ll multiply both sides
by two. And we get that 𝑦 is equal to two
𝑥. Replacing 𝑥 and 𝑦 with 𝑃𝑅 and
𝑂𝑅, respectively, we get that 𝑂𝑅 is equal to two multiplied by 𝑃𝑅. Another way of writing two
multiplied by 𝑃𝑅 is 𝑃𝑅 plus 𝑃𝑅.
However, we said earlier on that
the length 𝑃𝑅 is equal to the length 𝑄𝑅. Replacing one of the 𝑃𝑅s in this
expression with 𝑄𝑅 or 𝑅𝑄 to match the question gives us that 𝑂𝑅 is equal to
𝑃𝑅 plus 𝑅𝑄 as required.