# Video: Pack 5 β’ Paper 2 β’ Question 22

Pack 5 β’ Paper 2 β’ Question 22

03:24

### Video Transcript

The diagram shows a circle of center π and diameter π΄π΅. ππ· and π΄πΆ are parallel. The size of angle π΄π΅πΆ is 62 degrees. Find angle π΄πΆπ·.

Always say that whenever youβre doing this kind of question, the first thing you need to do is actually mark on any angles you can using the information that youβve got. Well, the first angle that we can mark on is actually the angle π΅πΆπ΄. And we can actually say that this is a right angle. And we can do that because π΄π΅πΆ is a triangle inscribed in a circle, where π΄π΅ is a diameter.

So therefore, π΅πΆπ΄ has got to be equal to 90 degrees. This is a relationship we know since itβs one of our circle theorems. The key thing here is that any value that you do find you need to give reasoning for. And thatβs what weβve done. We gave a reasoning for why itβs actually a right angle.

The next angle we can actually mark on is the angle π΅π΄πΆ, cause this is 28 degrees. And we get this because 180 minus 90 minus 62 equals 28. And we knew to use this calculation because the angles in a triangle actually add up to 180 degrees. So what weβve got is a triangle π΄π΅πΆ.

Well, weβve got the right angle, which is 90 degrees. Then weβre also given 62 degrees. Add these two together; we get 152 degrees, and 28 degrees will give us 180 degrees. So weβve actually calculated that and checked it. So we know that the angle π΅π΄πΆ is 28 degrees.

So we can also actually mark on angle π΄ππ· cause we can say that angle π΄ππ· is also gonna be 28 degrees. And we know that π΄ππ· is gonna be equal to 28 degrees because π΄ππ· and π΅π΄πΆ are alternate angles on parallel lines. This is sometimes known as Z angles, and Iβve actually shown this with a little sketch to the side.

Okay, great! But what else can we actually mark onto our diagram? Well, now what we can do is actually move on to the angle that we want to find in the question, and thatβs angle π΄πΆπ·. And we can actually do that using another relationship.

Well, the relationship that we have thatβs actually gonna help us to find this angle is this one here. So what we can see is that actually the angle subtended by an arc at the center is twice the angle subtended at the circumference. And Iβve actually shown this in a little sketch with π and two π. So we can actually see that the angle at the center of the circle is twice, because itβs two π, the angle actually on the circumference, which is π. But how does this apply to our diagram?

Well, we do actually have this relationship in our diagram. Iβve actually marked in pink on our diagram where this relationship is. And actually, it looks slightly different because itβs off center, and this can be actually what catches students out. So be careful of this because itβs such a common error.

So therefore, what we can say about our diagram is that π is the center of the circle and πΆ is on the circumference of the circle. So therefore, angle π΄ππ· is going to be twice angle π΄πΆπ·. And this actually comes from the circle theorem that we just looked at.

Okay, great! So letβs use this now to actually calculate what π΄πΆπ· is going to be. So therefore, angle π΄πΆπ· is gonna be equal to 28 divided by two because if π΄ππ· is twice π΄πΆπ·, then therefore angle π΄πΆπ· is going to be half of angle π΄ππ·. So therefore, we can say that angle π΄πΆπ· is gonna be equal to 14 degrees.