Video: AQA GCSE Mathematics Higher Tier Pack 2 β€’ Paper 1 β€’ Question 19

The diagram shows a kite and a rectangle. Given that π‘Ž is equal to 𝑏, write an equation for 𝑑 in terms of 𝑐.

03:40

Video Transcript

The diagram shows a kite and a rectangle. Given that π‘Ž is equal to 𝑏, write an equation for 𝑑 in terms of 𝑐.

It’s not always easy to spot how to answer a question like this. So I like to begin by recalling what I do know about the diagram. We have a kite and a rectangle. Remember we know that the interior angles in a rectangle are all 90 degrees. So this angle here must also be 90 degrees.

We also know that angles around a point sum to 360 degrees. So we can form an equation for the sum of the angles around this point. We can say that their sum is π‘Ž plus 𝑑 plus 90 plus 90. And we know that that must be equal to 360 degrees. 90 plus 90 is 180.

So this further simplifies to π‘Ž plus 𝑑 plus 180 equals 360. And we can subtract 180 from both sides of this equation to get π‘Ž plus 𝑑 is equal to 180. Now at this point, remember that we were told that π‘Ž is equal to 𝑏. So we can replace π‘Ž with 𝑏 in this equation to get 𝑏 plus 𝑑 is equal to 180.

And since we’re trying to form an equation for 𝑑, let’s make 𝑑 the subject by subtracting 𝑏 from both sides. And we can see that 𝑑 is equal to 180 minus 𝑏. Now, let’s go to the kite.

We know that these two opposite angles in the kite are equal in size. So this one here must also be 𝑏 degrees. And we have the fact that angles in a quadrilateral β€” a four-sided shape β€” sum to 360 degrees. So we can form an equation for the angles in the kite. Their sum is 90 plus 𝑏 plus 𝑏 plus 𝑐 and that is equal to 360 degrees. Since 𝑏 plus 𝑏 is two 𝑏, we can write that as 90 plus two 𝑏 plus 𝑐 is equal to 360.

Now, we’re trying to write an equation for 𝑑 in terms of 𝑐. So we’re going to need to rearrange this equation to make 𝑏 the subject. And this will give us an equation for 𝑏 in terms of 𝑐 that we can use later. To do this, we’ll first subtract 90 from both sides to get two 𝑏 minus 𝑐 is equal to 270.

We’ll then subtract 𝑐 which gives us two 𝑏 is equal to 270 minus 𝑐 and we’ll divide both sides by two. And our equation for 𝑏 in terms of 𝑐 is 270 minus 𝑐 over two. And now that we have an equation for 𝑏 in terms of 𝑐, we can replace the 𝑏 in our first equation with 270 minus 𝑐 all over two.

𝑑 is, therefore, equal to 180 minus 270 minus 𝑐 all over two. Now, we have got our equation for 𝑑 in terms of 𝑐. But it’s not particularly simple. What we’ll do to simplify is make the denominators of these two numbers the same.

We need to make 180 equal to 180 over one. And then, we need to multiply both the numerator and the denominator by two. That gives us 360 over two minus 270 minus 𝑐 over two. And then, we can subtract the numerators; that’s 360 minus 270 minus 𝑐 all over two.

Notice we put the 270 minus 𝑐 into brackets. You’ll see why in a moment. 360 minus 270 is 90. Then, we’re going to subtract a negative 𝑐. That’s the same as adding 𝑐. And we have 𝑑 is equal to 90 plus 𝑐 over two.

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