### Video Transcript

In the given figure, two tangents
𝑃𝑇 and 𝑃𝑆 are drawn, from an external point 𝑃, to a circle with center 𝑂 and
radius 𝑟. If 𝑂𝑃 is equal to two 𝑟, show
that the angle 𝑂𝑇𝑆 is equal to the angle 𝑂𝑆𝑇, which is equal to 30
degrees.

Let’s recall the circle theorems we
know that might be useful here. Firstly, we know that the radius
and the tangent meet at an angle of 90 degrees. This means that triangle 𝑂𝑇𝑃 is
right angled at 𝑇. Let’s call the angle 𝑇𝑂𝑃 𝜃 for
now. We’ll use right angle trigonometry
to calculate the size of this angle.

The line joining 𝑂 to the tangent
at 𝑇 must be the radius. So we can call that 𝑟. And we’re told that the line 𝑂𝑃
is equal to two 𝑟. We can label this triangle with
respect to the angle 𝜃. The longest side of the triangle is
the line 𝑂𝑃. That’s the one opposite the right
angle. We call that the hypotenuse. The line 𝑇𝑃 is directly opposite
the angle 𝜃. So that’s the opposite. The remaining side 𝑂𝑇 is the
adjacent. That’s the one next to the angle
𝜃.

Since we have expressions for the
length of the adjacent and the hypotenuse, we can use the cosine ratio: cos of 𝜃 is
equal to adjacent over hypotenuse. Substituting what we know into this
formula and we get cos of 𝜃 is equal to 𝑟 over two 𝑟. The 𝑟s cancel and we’re left with
cos of 𝜃 is equal to one-half. We know, however, that cos of 60 is
equal to one-half. So we’ve calculated that the angle
that we labelled 𝜃 is 60 degrees.

Let’s repeat this process for the
triangle 𝑂𝑃𝑆. Let’s call our angle this time
𝛼. We can label this triangle as
shown. In fact, substituting what we know
about triangle 𝑂𝑃𝑆 into our formula for the cosine ratio and we get an almost
identical expression to the one that we did for 𝑂𝑇𝑃. 𝛼 is also 60 degrees.

Now, let’s consider triangle
𝑂𝑇𝑆. We know that both 𝑂𝑇 and 𝑂𝑆 are
radii of the circle center 𝑂. This means that the lines 𝑂𝑇 and
𝑂𝑆 are of equal length. And triangle 𝑂𝑇𝑆 is
isosceles.

Angles in a triangle add to 180
degrees. So we can calculate the total
measure of the angles 𝑂𝑇𝑆 and 𝑂𝑆𝑇 by subtracting 60 and 60 from 180. That gives us 60 degrees. Since we know that two angles in an
isosceles triangle are equal, we can divide 60 by two to work out the size of angle
𝑂𝑇𝑆 and 𝑂𝑆𝑇. 60 divided by two is 30.

We have shown as required the angle
𝑂𝑇𝑆 is equal to angle 𝑂𝑆𝑇, which is equal to 30 degrees.