# Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 21

𝐴, 𝐵, and 𝐶 are points on a circle and 𝑂 is the center of the circle. 𝐷𝐶𝐸 is a tangent to the circle. 𝐶𝐵 bisects the angle 𝐴𝐶𝐸. Angle 𝐵𝐶𝐸 = 𝑥° and angle 𝑂𝐵𝐶 = (90 − 𝑥°). Prove that 𝐴𝐵 = 𝐵𝐶.

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### Video Transcript

𝐴, 𝐵, and 𝐶 are points on a circle and 𝑂 is the center of the circle. 𝐷𝐶𝐸 is a tangent to the circle. 𝐶𝐵 bisects the angle 𝐴𝐶𝐸. Angle 𝐵𝐶𝐸 is equal to 𝑥 degrees and angle 𝑂𝐵𝐶 is equal to 90 minus 𝑥 degrees. Prove that 𝐴𝐵 is equal to 𝐵𝐶.

In order to prove that 𝐴𝐵 is equal to 𝐵𝐶, we need to show that triangle 𝐴𝐵𝐶 is isosceles. If this is the case, it will have two equal angles at 𝐵𝐴𝐶 and 𝐵𝐶𝐴. And this will mean the sides 𝐴𝐵 and 𝐵𝐶 have to be equal in length. Let’s first see if there’s any useful information that we can use in the question itself.

We’re told that 𝐶𝐵 bisects the angle 𝐴𝐶𝐸; that is to say, the line 𝐶𝐵 cuts the angle 𝐴𝐶𝐸 in half. For this to be the case, that must mean that angle 𝐵𝐶𝐸 is equal to angle 𝐵𝐶𝐴. They’re both 𝑥 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. 𝑂𝐵 and 𝑂𝐶 are lines which join the center of the circle to a point on its circumference. That means they’re the radii of the circle and triangle 𝑂𝐵𝐶 is isosceles.

This means we know the 𝑂𝐶𝐵 and 𝑂𝐵𝐶 are equal. They’re both 90 minus 𝑥 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 𝐵𝑂𝐶 marked. Angle 𝐵𝑂𝐶 can be found by subtracting angle 𝑂𝐵𝐶 and angle 𝑂𝐶𝐵 from 180 degrees. 180 minus 90 minus 90 is zero. Negative negative 𝑥 is simply 𝑥, minus negative 𝑥 is plus 𝑥. And we can see that angle 𝐵𝑂𝐶 is equal to two 𝑥 degrees.

We can see now that angle 𝐵𝑂𝐶 and 𝐵𝐴𝐶 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 𝐵𝐴𝐶 is half angle 𝐵𝑂𝐶. It’s a half of two 𝑥 which is simply 𝑥 degrees.

We have now shown that 𝐴𝐵𝐶 is an isosceles triangle with equal angles at 𝐵𝐴𝐶 and 𝐵𝐶𝐴. This means in turn that the sides 𝐴𝐵 and 𝐵𝐶 must also be of equal length. We have proven that 𝐴𝐵 is equal to 𝐵𝐶.