𝐴𝐶𝐷𝐹 is a rectangle. 𝐴𝐵𝐸𝐹 is a square. The diagonal 𝐴𝐷 intersects 𝐵𝐸
at 𝐺. The area of triangle 𝐴𝐵𝐺 is five
centimetres squared. The area of triangle 𝐵𝐶𝐺 is 15
centimetres squared. Find the area of 𝐴𝐶𝐷𝐹.
Well, it says that 𝐴𝐶𝐷𝐹 is a
rectangle. So we know that we’ve got four
right angles, and opposite sides are equal in length. And 𝐴𝐵𝐸𝐹 is a square. So again, we’ve got four right
angles, and all of the sides are the same length. Also, because 𝐴𝐵𝐶 is a straight
line and 𝐴𝐵𝐸 is a right angle, we know that 𝐶𝐵𝐺 is also a right angle.
Now we can also see that lines
𝐴𝐹, 𝐵𝐸, and 𝐶𝐷 are parallel, as are lines 𝐴𝐵𝐶 and 𝐹𝐸𝐷. The question also tells us the area
of triangle 𝐴𝐵𝐺 is five square centimetres and the area of triangle 𝐵𝐶𝐺 is 15
square centimetres. So we’ve been given the areas of
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 𝐴𝐵 the base of
triangle 𝐴𝐵𝐺 and 𝐵𝐺 the perpendicular height, the area of triangle 𝐴𝐵𝐺 can
also be written as a half times the length 𝐴𝐵 times the length of 𝐵𝐺. And of course, we know that that’s
five square centimetres. And for triangle 𝐵𝐶𝐺, we’ll take
𝐵𝐶 as the base and again 𝐵𝐺 as the perpendicular height. So we’ve got a half times length
𝐵𝐶 times length 𝐵𝐺 is equal to 15 square centimetres in this case.
Now if we look at these two, we can
see that they’re both a half times something, and they both got 𝐵𝐺 as the
perpendicular height. So the only thing that differs is
the length of the base, 𝐴𝐵 and 𝐵𝐶. 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 𝐵𝐶 must be
three times bigger than 𝐴𝐵 in order for that to work.
Now just to make our writing a
little bit more efficient, we’re going to define a new variable called 𝑎, which is
the length of 𝐴𝐵. Now you don’t have to do this, but
it does make the writing little bit easier. So on our diagram, this distance
here is 𝑎, which means that this distance here between 𝐵 and 𝐶 is three times
that, three 𝑎.
Now remember, 𝐴𝐵𝐸𝐹 was a
square, so 𝐴𝐹 has got length 𝑎 and 𝐹𝐸 has also got length 𝑎. And because 𝐴𝐶𝐷𝐹 is a
rectangle, that means that 𝐶𝐷 must have length 𝑎 and 𝐸𝐷 must have length three
𝑎 as well. And the overall length of that
rectangle from 𝐹 to 𝐷 or 𝐴 to 𝐶 is 𝑎 plus three 𝑎, which is four 𝑎.
Now let’s think about triangles
𝐴𝐵𝐺 and 𝐴𝐶𝐷. Well, they’re both right-angled
triangles. And this angle here is this angle
here and this angle here, so they’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 𝐴𝐵𝐺
and 𝐴𝐶𝐷 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 𝐴𝐵𝐺 are a quarter of the length of the corresponding sides in
𝐴𝐶𝐷 because length 𝐴𝐵 is a quarter of the length of 𝐴𝐶. And this means that length 𝐵𝐺 is
a quarter of the length of 𝐶𝐷. And since we know that length 𝐶𝐷
is equal to this letter 𝑎 that we defined, we know that 𝐵𝐺 has a length 𝑎 over
four, or a quarter 𝑎.
Now we know the perpendicular
height of triangle 𝐴𝐵𝐺. We can plug that into the formula
for the area and find out the value of 𝑎. And in our case, the area was
five. So the area of five is equal to a
half times the base length, 𝑎, times the perpendicular height length, 𝑎 over
Now it’s probably easier to think
of 𝑎 as being 𝑎 over one in this case, so we’ve got three fractions multiplied
together. So on the numerator, I’ve got one
times 𝑎 times 𝑎, which is 𝑎 squared. And on the denominator, I’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 𝑎 squared is
equal to 40. And that means that 𝑎 is equal to
the square root of 40. Well, it could be the positive or
negative version of that, mathematically speaking. But because we’re dealing with
length, we’re not interested in the negative version.
But before we worry about trying to
work out what the square root of 40 is, let’s think what we’re trying to
achieve. We’re trying to find the area of
rectangle 𝐴𝐶𝐷𝐹. And to do that, we’re gonna do the
base times the height.
Now the base length here is four
𝑎, and the height is 𝑎. So that area, four 𝑎 times 𝑎, is
four 𝑎 squared. But we know that 𝑎 squared is
equal to 40, so we can just plug that number in for 𝑎 squared here. And four times 40 is 160. So without forgetting the units,
the area of rectangle 𝐴𝐶𝐷𝐹 is 160 square centimetres.
So I’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.