# Video: Pack 4 β’ Paper 3 β’ Question 20

Pack 4 β’ Paper 3 β’ Question 20

03:36

### Video Transcript

Shown is a circle with centre π. π΄, π΅, and πΆ lie on the circumference of the circle. Prove that π¦ is equal to two π₯. Do not use any circle theorems in your proof.

This is an example of a geometric proof. In order to prove the statement that π¦ is equal to two π₯, we need to use a series of known facts. Weβve been told that weβre not allowed to use any circle theorems. Therefore, we will need to use the angle facts we know.

Here, we have two isosceles triangles. The first is triangle π΄ππ΅ we know that itβs an isosceles triangle because the sides π΄π and ππ΅ as shown are the radii of the circle, which means they have to be of the same length. Since this is an isosceles triangle, we know that angle ππ΄π΅ and ππ΅π΄ are the same size. We will call these π for now.

Next, letβs use the fact that angles in a triangle add to 180 degrees. We can find the remaining angle in this triangle π΄ππ΅ by subtracting the known angles from 180. Letβs call this angle π. π is equal to 180 minus π plus π, which can be simplified to 180 minus two π.

Next, we can repeat the process for the other isosceles triangle π΅ππΆ. Weβll call these two equal angles π this time. Once again, since angles in a triangle add to 180 degrees, the angle π΅ππΆ which weβve now called π is 180 minus π plus π or 180 minus two π.

The final fact that we need to use is that angles around a point add to 360 degrees. This helps us find an expression for the angle labelled π¦. π¦ is equal to 360 minus π minus π. Remember we said that π was equal to 180 minus two π and π was equal to 180 minus two π. We can substitute these expressions that we originally created for π and π into our equation for π¦.

Next, weβll expand the brackets really carefully. Remember since each bracket has a minus sign in front of it, itβs technically being multiplied by negative one. That gives us 360 minus 180 plus two π minus 180 plus two π. Now, 360 minus 180 minus 180 is zero. So our equation for π¦ is π¦ is equal to two π plus two π. And we can factorize this to get two lots of π plus π.

But letβs look back to our diagram. Angle π and π combine to make angle π₯. So π₯ is equal to π plus π. In this case then, we will replace π plus π with π₯ and we get π¦ is equal to two π₯. That is what weβre being asked to prove. So we are done.

Remember when weβre performing a geometric proof, we must ensure that we explicitly write down all the angle rules that weβve used. Here that was rules about angles in a triangle adding to 180 degrees, angles in an isosceles triangle, and angles around a point adding to 360 degrees.