The diagram shows a kite and a
rectangle. Given that 𝑎 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
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 𝑎
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 𝑎
plus 𝑑 plus 180 equals 360. And we can subtract 180 from both
sides of this equation to get 𝑎 plus 𝑑 is equal to 180. Now at this point, remember that we
were told that 𝑎 is equal to 𝑏. So we can replace 𝑎 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 — a four-sided shape — 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.