Video: AQA GCSE Mathematics Higher Tier Pack 5 β€’ Paper 3 β€’ Question 15

Points 𝐴, 𝐡, and 𝐢 are on the circumference of the circle. Points 𝐷 and 𝐸 are inside the circle. Circle the lines which are radii of the circle. [A] 𝐡𝐷 [B] 𝐡𝐸 [C] 𝐢𝐷 [D] 𝐢𝐸.

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Video Transcript

Points 𝐴, 𝐡, and 𝐢 are on the circumference of the circle. Points 𝐷 and 𝐸 are inside the circle. Circle the lines which are radii of the circle, 𝐡𝐷, 𝐡𝐸, 𝐢𝐷, 𝐢𝐸. And it’s important to notice that this circle is not drawn accurately.

Radii is plural for radius. And the radius of the circle is a line that joins the point at the center of the circle to a point on its circumference. In order to answer this question, we’re going to need to recall circle theorems.

You might have spotted that arrowhead shape. Now, this circle theorem says that the angle at the center is double the angle at the circumference. And this is sometimes written that the angle subtended by an arc at the center is twice the angle subtended at the circumference. On this general diagram, these two lines marked must be the radii. So let’s use this information to help us decide which lines on our circle are the radii of the circle.

We can see that the angle at the circumference of our circle is 70 degrees. This means the angle subtended at the center must be 140 degrees. So we need to decide whether angle 𝐡𝐷𝐢 or 𝐡𝐸𝐢 is 140 degrees. Remember, angles around a point sum to 360 degrees. So we can work out the measure of the angle at 𝐡𝐷𝐢 by subtracting 290 degrees from 360 degrees. That’s 70 degrees. So angle 𝐡𝐷𝐢 is 70 degrees. Similarly, we can find the measure of the obtuse angle at 𝐡𝐸𝐢 by subtracting 220 from 360 degrees, which is 140 degrees. Remember, we said the angle at the center had to be 140 degrees. This must mean that 𝐸 is the center of the circle.

And since the radius is a line joining the point at the center of the circle to a point on its circumference, these are our two radii. They are line 𝐡𝐸 and line 𝐢𝐸.

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