Video: AQA GCSE Mathematics Higher Tier Pack 5 โ€ข Paper 2 โ€ข Question 10

๐ด, ๐ต, and ๐ถ are points on a circle. ๐ต๐ท is a tangent to the circle. Write down the size of angle ๐‘ฅ. Give a reason for your answer.

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Video Transcript

๐ด, ๐ต, and ๐ถ are points on a circle. ๐ต๐ท is a tangent to the circle. Write down the size of angle ๐‘ฅ. Give a reason for your answer.

This is a circle theorem question. And there is a little bit of a clue about how much work we actually need to do to answer it in the question. The words โ€œwrite downโ€ indicate that thereโ€™s little to no working out we need to do. In fact, we should be able to work out the size of angle ๐‘ฅ just by looking at our diagram.

There are a few unnecessary bits of information. We actually donโ€™t need to worry about the angles ๐ถ๐ด๐ต and ๐ด๐ต๐ถ. The only one weโ€™re interested in is this one at ๐ด๐ถ๐ต. Our angle ๐‘ฅ is actually the same as this angle. Itโ€™s 80 degrees. And this is called the alternate segment theorem. This only works when we have a triangle inscribed in a circle with a tangent at one of its vertices.

So in this example, these two angles would be equal as with these two. Using the same rule, we can also see that this angle here is the same as angle ๐ถ๐ด๐ต. Itโ€™s 65 degrees. ๐‘ฅ is 80 degrees. And it is enough to simply say the reason is because of the alternate segment theorem.

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