Video: Pack 5 β€’ Paper 2 β€’ Question 22

Pack 5 β€’ Paper 2 β€’ Question 22


Video Transcript

The diagram shows a circle of center 𝑀 and diameter 𝐴𝐡. 𝑀𝐷 and 𝐴𝐢 are parallel. The size of angle 𝐴𝐡𝐢 is 62 degrees. Find angle 𝐴𝐢𝐷.

Always say that whenever you’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 you’ve got. Well, the first angle that we can mark on is actually the angle 𝐡𝐢𝐴. And we can actually say that this is a right angle. And we can do that because 𝐴𝐡𝐢 is a triangle inscribed in a circle, where 𝐴𝐡 is a diameter.

So therefore, 𝐡𝐢𝐴 has got to be equal to 90 degrees. This is a relationship we know since it’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 that’s what we’ve done. We gave a reasoning for why it’s actually a right angle.

The next angle we can actually mark on is the angle 𝐡𝐴𝐢, 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 we’ve got is a triangle 𝐴𝐡𝐢.

Well, we’ve got the right angle, which is 90 degrees. Then we’re also given 62 degrees. Add these two together; we get 152 degrees, and 28 degrees will give us 180 degrees. So we’ve actually calculated that and checked it. So we know that the angle 𝐡𝐴𝐢 is 28 degrees.

So we can also actually mark on angle 𝐴𝑀𝐷 cause we can say that angle 𝐴𝑀𝐷 is also gonna be 28 degrees. And we know that 𝐴𝑀𝐷 is gonna be equal to 28 degrees because 𝐴𝑀𝐷 and 𝐡𝐴𝐢 are alternate angles on parallel lines. This is sometimes known as Z angles, and I’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 that’s angle 𝐴𝐢𝐷. And we can actually do that using another relationship.

Well, the relationship that we have that’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 I’ve actually shown this in a little sketch with πœƒ and two πœƒ. So we can actually see that the angle at the center of the circle is twice, because it’s two πœƒ, the angle actually on the circumference, which is πœƒ. But how does this apply to our diagram?

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

So therefore, what we can say about our diagram is that 𝑀 is the center of the circle and 𝐢 is on the circumference of the circle. So therefore, angle 𝐴𝑀𝐷 is going to be twice angle 𝐴𝐢𝐷. And this actually comes from the circle theorem that we just looked at.

Okay, great! So let’s use this now to actually calculate what 𝐴𝐢𝐷 is going to be. So therefore, angle 𝐴𝐢𝐷 is gonna be equal to 28 divided by two because if 𝐴𝑀𝐷 is twice 𝐴𝐢𝐷, then therefore angle 𝐴𝐢𝐷 is going to be half of angle 𝐴𝑀𝐷. So therefore, we can say that angle 𝐴𝐢𝐷 is gonna be equal to 14 degrees.

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