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

π΄, π΅, πΆ, and π· are points on a circle with center π. π΄πΆ is a diameter. πΆπ 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 π¦.

Letβs start with the first sentence. π΄, π΅, πΆ, and π· are points on a circle with center π. Hereβs our center π. Points π΄, π΅, πΆ, and π· fall on the circumference of the circle and cross through the center. That means that π΄π is a radius, ππ΅ is a radius, ππΆ is a radius, and ππ· is a radius. This means that triangle ππ΅πΆ is isosceles.

Angle ππ΅πΆ is equal to the angle of ππΆπ΅. Angle π΅ππΆ 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. ππ΅πΆ equals 35 degrees and ππΆπ΅ equals 35 degrees.

Weβre told that line πΆπ is a tangent to the circle. And that means that the angle ππΆπ 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, letβs look at the place that line ππΆ and line π΅π· intersect. Weβll call this intersection point πΈ. Angle ππΈπ΅ equals 58 degrees. This is because vertically opposite angles are equivalent. Because π΅π· is a straight line, angle ππΈπ· must be equal to 180 degrees minus 58 degrees. Angle ππΈπ· then measures 122 degrees.

We still have a few more angles we need to find before we can say what π¦ is. Looking at the triangle created from points ππΈπ΅ here in yellow, if we add 110 degrees plus 58 degrees and then subtract that from 180, weβll find the angle ππ΅πΈ. 110 degrees plus 58 degrees equals 168 degrees. When we subtract that from 180, we get 12 degrees. Angle ππ΅πΈ equals 12 degrees. We know that triangle π·ππ΅ is isosceles. ππ΅ and ππ· are radii. And that means angle ππ·π΅ is equal to angle ππ΅π·. And they both are 12 degrees.

Finally, letβs consider triangle ππΈπ·. The three angles inside triangle ππΈπ· must add up to 180 degrees. One of the angles is π¦. One of the angles is 122 degrees. And the last angle is 12 degrees. 180 degrees minus 134 degrees equals π¦. π¦ equals 46 degrees.