Video: AQA GCSE Mathematics Higher Tier Pack 3 β€’ Paper 2 β€’ Question 24

In the diagram, 𝐴𝐡𝐢𝐷 is a kite, and 𝐡𝐸𝐹𝐢 and 𝐢𝐺𝐻𝐷 are squares. Prove that π‘₯ = 3𝑦.

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Video Transcript

In the diagram, 𝐴𝐡𝐢𝐷 is a kite, and 𝐡𝐸𝐹𝐢 and 𝐢𝐺𝐻𝐷 are squares. Prove that π‘₯ is equal to three 𝑦.

So our goal is to prove that π‘₯ is equal to three 𝑦. And we have two 𝑦 and 𝑦 in our diagram, along with our angle π‘₯. So to prove this, let’s first go through the given information and try to use properties that might be useful to prove this.

So first, we know that 𝐴𝐡𝐢𝐷 is a kite. So if we have a kite, we can say that angle 𝐡𝐴𝐷 is equal to angle 𝐡𝐢𝐷 because it’s a kite. In a kite, the angles across from each other are congruent. So like we said, angle 𝐡𝐴𝐷 and angle 𝐡𝐢𝐷 would be equal. And since angle 𝐡𝐴𝐷 is equal to π‘₯, we can also call angle 𝐡𝐢𝐷 π‘₯.

Next, we are given two squares: 𝐡𝐸𝐹𝐢 and 𝐢𝐺𝐻𝐷. Now instead of using every property of a square and every property of a kite to label this diagram, we’re trying to prove that π‘₯ is equal to three 𝑦 and all of the π‘₯s and 𝑦s that are found in our diagram are angles. So let’s go through the angle information that we could include on our diagram.

So since we have these two squares, in a square each of the angles are equal to 90 degrees. So as we said, each of these angles would be equal to 90 degrees. So they’re all equal to each other and each of them will be 90 degrees because 𝐡𝐸𝐹𝐢 is a square. And because 𝐢𝐺𝐻𝐷 is a square, each of its angles are equal to 90 degrees.

So the only shape that we haven’t touched is triangle 𝐢𝐹𝐺. But we were given that angle 𝐢𝐹𝐺 was equal to two 𝑦 and angle 𝐹𝐺𝐢 is equal to 𝑦. And all angles in a triangle sum to 180 degrees. So this missing angle 𝐹𝐢𝐺, we could say, we take 180 degrees and subtract the other two angles from that. That would be what’s left for angle 𝐹𝐢𝐺. So angle 𝐹𝐢𝐺 will be equal to 180 minus two 𝑦 and the 𝑦. And simplifying, we could say angle 𝐹𝐢𝐺 will be equal to 180 minus three 𝑦.

Alright, now that we’ve labelled on pretty much every angle in our diagram, we’re trying to prove that π‘₯ is equal to three 𝑦 and we have an π‘₯ in the diagram that’s pink and an π‘₯ in the diagram that’s blue. Now, the π‘₯ in the diagram that is blue is located near a point with angles around that point. And angles around a point sum to 360 degrees. So if this is our point, all of the angles around this would sum to 360 degrees.

So if we wanted to come up with an equation that involved π‘₯, we could say that π‘₯ is equal to 360 minus those other three angles. So we take 360 and we need to subtract 90, 180 minus three 𝑦, and 90. And again, this is because angles around a point sum to 360 degrees. So let’s begin simplifying.

Inside our brackets, 90 plus 180 plus 90 is 360. But we can’t forget about the minus three 𝑦. Now, expanding this minus sign, we have that π‘₯ is equal to 360 minus 360 plus three 𝑦. So the 360s cancel. And we’re left with π‘₯ is equal to three 𝑦, which is exactly what we were trying to prove so we would be finished.

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