Video: CBSE Class X • Pack 3 • 2016 • Question 10

CBSE Class X • Pack 3 • 2016 • Question 10


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.

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