# Video: Pack 1 β’ Paper 1 β’ Question 21

Pack 1 β’ Paper 1 β’ Question 21

06:14

### Video Transcript

π΄πΆπ·πΉ is a rectangle. π΄π΅πΈπΉ is a square. The diagonal π΄π· intersects π΅πΈ at πΊ. The area of triangle π΄π΅πΊ is five centimetres squared. The area of triangle π΅πΆπΊ is 15 centimetres squared. Find the area of π΄πΆπ·πΉ.

Well, it says that π΄πΆπ·πΉ is a rectangle. So we know that weβve got four right angles, and opposite sides are equal in length. And π΄π΅πΈπΉ is a square. So again, weβve got four right angles, and all of the sides are the same length. Also, because π΄π΅πΆ is a straight line and π΄π΅πΈ is a right angle, we know that πΆπ΅πΊ is also a right angle.

Now we can also see that lines π΄πΉ, π΅πΈ, and πΆπ· are parallel, as are lines π΄π΅πΆ and πΉπΈπ·. The question also tells us the area of triangle π΄π΅πΊ is five square centimetres and the area of triangle π΅πΆπΊ is 15 square centimetres. So weβve been given the areas of some triangles.

Now if you remember, the area of a triangle is equal to a half times its base times its perpendicular height. So if we call π΄π΅ the base of triangle π΄π΅πΊ and π΅πΊ the perpendicular height, the area of triangle π΄π΅πΊ can also be written as a half times the length π΄π΅ times the length of π΅πΊ. And of course, we know that thatβs five square centimetres. And for triangle π΅πΆπΊ, weβll take π΅πΆ as the base and again π΅πΊ as the perpendicular height. So weβve got a half times length π΅πΆ times length π΅πΊ is equal to 15 square centimetres in this case.

Now if we look at these two, we can see that theyβre both a half times something, and they both got π΅πΊ as the perpendicular height. So the only thing that differs is the length of the base, π΄π΅ and π΅πΆ. And we can also see that the area of the second triangle is three times bigger than the area of the first triangle. So we can see that π΅πΆ must be three times bigger than π΄π΅ in order for that to work.

Now just to make our writing a little bit more efficient, weβre going to define a new variable called π, which is the length of π΄π΅. Now you donβt have to do this, but it does make the writing little bit easier. So on our diagram, this distance here is π, which means that this distance here between π΅ and πΆ is three times that, three π.

Now remember, π΄π΅πΈπΉ was a square, so π΄πΉ has got length π and πΉπΈ has also got length π. And because π΄πΆπ·πΉ is a rectangle, that means that πΆπ· must have length π and πΈπ· must have length three π as well. And the overall length of that rectangle from πΉ to π· or π΄ to πΆ is π plus three π, which is four π.

Now letβs think about triangles π΄π΅πΊ and π΄πΆπ·. Well, theyβre both right-angled triangles. And this angle here is this angle here and this angle here, so theyβve got that angle in common. Now angles in a triangle sum to 180 degrees. So that means that this angle must be the same as this angle, because the other two angles are the same. This means that triangles π΄π΅πΊ and π΄πΆπ· are similar.

In similar triangles, all the sides are in the same ratios, and the sides of one triangle are a simple multiple of the sides of the other triangle. We can see that the corresponding sides in triangle π΄π΅πΊ are a quarter of the length of the corresponding sides in π΄πΆπ· because length π΄π΅ is a quarter of the length of π΄πΆ. And this means that length π΅πΊ is a quarter of the length of πΆπ·. And since we know that length πΆπ· is equal to this letter π that we defined, we know that π΅πΊ has a length π over four, or a quarter π.

Now we know the perpendicular height of triangle π΄π΅πΊ. We can plug that into the formula for the area and find out the value of π. And in our case, the area was five. So the area of five is equal to a half times the base length, π, times the perpendicular height length, π over four.

Now itβs probably easier to think of π as being π over one in this case, so weβve got three fractions multiplied together. So on the numerator, Iβve got one times π times π, which is π squared. And on the denominator, Iβve got two times one times four, which is eight.

Now I can multiply both sides of my equation by eight so that the eights cancel on the right-hand side. And that gives me π squared is equal to 40. And that means that π is equal to the square root of 40. Well, it could be the positive or negative version of that, mathematically speaking. But because weβre dealing with length, weβre not interested in the negative version.

But before we worry about trying to work out what the square root of 40 is, letβs think what weβre trying to achieve. Weβre trying to find the area of rectangle π΄πΆπ·πΉ. And to do that, weβre gonna do the base times the height.

Now the base length here is four π, and the height is π. So that area, four π times π, is four π squared. But we know that π squared is equal to 40, so we can just plug that number in for π squared here. And four times 40 is 160. So without forgetting the units, the area of rectangle π΄πΆπ·πΉ is 160 square centimetres.

So Iβve ruled there was quite a lot to do in this question. We had to know the properties of rectangles and squares. We had to remember the formula for the area of a triangle. And we also had to recognise some similar triangles and use that information to work out the length of various sides and then use that to work out the area of the final rectangle.

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