# Video: AQA GCSE Mathematics Higher Tier Pack 5 β’ Paper 1 β’ Question 21

π΄, π΅, and πΆ are points on a circle and π is the center of the circle. π·πΆπΈ is a tangent to the circle. πΆπ΅ bisects the angle π΄πΆπΈ. Angle π΅πΆπΈ = π₯Β° and angle ππ΅πΆ = (90 β π₯Β°). Prove that π΄π΅ = π΅πΆ.

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### Video Transcript

π΄, π΅, and πΆ are points on a circle and π is the center of the circle. π·πΆπΈ is a tangent to the circle. πΆπ΅ bisects the angle π΄πΆπΈ. Angle π΅πΆπΈ is equal to π₯ degrees and angle ππ΅πΆ is equal to 90 minus π₯ degrees. Prove that π΄π΅ is equal to π΅πΆ.

In order to prove that π΄π΅ is equal to π΅πΆ, we need to show that triangle π΄π΅πΆ is isosceles. If this is the case, it will have two equal angles at π΅π΄πΆ and π΅πΆπ΄. And this will mean the sides π΄π΅ and π΅πΆ have to be equal in length. Letβs first see if thereβs any useful information that we can use in the question itself.

Weβre told that πΆπ΅ bisects the angle π΄πΆπΈ; that is to say, the line πΆπ΅ cuts the angle π΄πΆπΈ in half. For this to be the case, that must mean that angle π΅πΆπΈ is equal to angle π΅πΆπ΄. Theyβre both π₯ degrees. Next, we need to recall any circle theorems that might help us here. Straightaway, I can see one that might be useful. We know that two radii or radiuses form an isosceles triangle. ππ΅ and ππΆ are lines which join the center of the circle to a point on its circumference. That means theyβre the radii of the circle and triangle ππ΅πΆ is isosceles.

This means we know the ππΆπ΅ and ππ΅πΆ are equal. Theyβre both 90 minus π₯ degrees. We also know that angles in a triangle sum to 180 degrees. So we can use this information to calculate the size of angle π΅ππΆ marked. Angle π΅ππΆ can be found by subtracting angle ππ΅πΆ and angle ππΆπ΅ from 180 degrees. 180 minus 90 minus 90 is zero. Negative negative π₯ is simply π₯, minus negative π₯ is plus π₯. And we can see that angle π΅ππΆ is equal to two π₯ degrees.

We can see now that angle π΅ππΆ and π΅π΄πΆ are subtended from points on the circumference of the circle. We can, therefore, use the circle theorem that says the angle at the center is twice the angle at the circumference. Another way to think about this is that the angle at the circumference is half the angle at the center. Angle π΅π΄πΆ is half angle π΅ππΆ. Itβs a half of two π₯ which is simply π₯ degrees.

We have now shown that π΄π΅πΆ is an isosceles triangle with equal angles at π΅π΄πΆ and π΅πΆπ΄. This means in turn that the sides π΄π΅ and π΅πΆ must also be of equal length. We have proven that π΄π΅ is equal to π΅πΆ.