Video: AQA GCSE Mathematics Higher Tier Pack 1 β€’ Paper 1 β€’ Question 21

AQA GCSE Mathematics Higher Tier Pack 1 β€’ Paper 1 β€’ Question 21

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Video Transcript

𝐴, 𝐡, 𝐢, and 𝐷 are points on a circle with center 𝑂. 𝐴𝐢 is a diameter. 𝐢𝑇 is a tangent to the circle. This figure is not drawn to scale. Work out the size of angle π‘₯ and the size of angle 𝑦.

Let’s start with the first sentence. 𝐴, 𝐡, 𝐢, and 𝐷 are points on a circle with center 𝑂. Here’s our center 𝑂. Points 𝐴, 𝐡, 𝐢, and 𝐷 fall on the circumference of the circle and cross through the center. That means that 𝐴𝑂 is a radius, 𝑂𝐡 is a radius, 𝑂𝐢 is a radius, and 𝑂𝐷 is a radius. This means that triangle 𝑂𝐡𝐢 is isosceles.

Angle 𝑂𝐡𝐢 is equal to the angle of 𝑂𝐢𝐡. Angle 𝐡𝑂𝐢 equals 110 degrees. If we subtract 110 degrees from 180 degrees, we get 70 degrees. We know that this 70 degrees must be divided evenly into the two remaining angles. 70 degrees divided by two equals 35 degrees. 𝑂𝐡𝐢 equals 35 degrees and 𝑂𝐢𝐡 equals 35 degrees.

We’re told that line 𝐢𝑇 is a tangent to the circle. And that means that the angle 𝑂𝐢𝑇 is 90 degrees. A radius meets a tangent line of a circle at 90 degrees. The size of angle π‘₯ is 90 degrees minus 35 degrees. The size of angle π‘₯ is 55 degrees.

Next, let’s look at the place that line 𝑂𝐢 and line 𝐡𝐷 intersect. We’ll call this intersection point 𝐸. Angle 𝑂𝐸𝐡 equals 58 degrees. This is because vertically opposite angles are equivalent. Because 𝐡𝐷 is a straight line, angle 𝑂𝐸𝐷 must be equal to 180 degrees minus 58 degrees. Angle 𝑂𝐸𝐷 then measures 122 degrees.

We still have a few more angles we need to find before we can say what 𝑦 is. Looking at the triangle created from points 𝑂𝐸𝐡 here in yellow, if we add 110 degrees plus 58 degrees and then subtract that from 180, we’ll find the angle 𝑂𝐡𝐸. 110 degrees plus 58 degrees equals 168 degrees. When we subtract that from 180, we get 12 degrees. Angle 𝑂𝐡𝐸 equals 12 degrees. We know that triangle 𝐷𝑂𝐡 is isosceles. 𝑂𝐡 and 𝑂𝐷 are radii. And that means angle 𝑂𝐷𝐡 is equal to angle 𝑂𝐡𝐷. And they both are 12 degrees.

Finally, let’s consider triangle 𝑂𝐸𝐷. The three angles inside triangle 𝑂𝐸𝐷 must add up to 180 degrees. One of the angles is 𝑦. One of the angles is 122 degrees. And the last angle is 12 degrees. 180 degrees minus 134 degrees equals 𝑦. 𝑦 equals 46 degrees.

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