The shape 𝐴𝐶𝐷 is a triangle. The two marked angles 𝐴𝐸𝐵 and
𝐴𝐷𝐶 are equal. 𝐴𝐵 is equal to 6.8
centimeters. 𝐴𝐶 is equal to 8.5
centimeters. 𝐷𝐶 is equal to 9.5 centimeters.
a) Find the length of 𝐸𝐵 and b) find the length of 𝐴𝐷.
Well, the first thing to know is
that the diagram is not drawn to scale. So therefore, we can’t just measure
the sides and answer questions that way. We’re gonna have to use another
method. So the first step I’m actually
gonna complete is to mark on all the information that we know.
So first of all, what I’ve actually
done is marked on the lengths of the sides we know. So we know that 𝐷𝐶 is equal to
9.5, 𝐴𝐵 is equal to 6.8, and 𝐴𝐶 is equal to 8.5 centimeters. What we also know is that we got
angle 𝐷𝐴𝐶 in both triangles. So therefore, we got angle 𝐷𝐴𝐶
in both triangles. Then, we know that our left-hand
angles are the same in both triangles. So therefore, angle 𝐴𝐵𝐸 must be
equal to angle 𝐴𝐶𝐷. And also, this is because they’re
corresponding angles because we’ve now got a pair of parallel lines.
Okay, great, so we’ve now got all
the information on the triangles that we need. So let’s get on and find the length
of 𝐸𝐵. So the first important thing to
note is that our triangles 𝐴𝐵𝐸 and 𝐴𝐶𝐷 are similar. And they’re similar because
actually they’ve got three angles that are the same because we’ve got the angle
𝐷𝐴𝐶, which is shared. Then, we know that 𝐴𝐸𝐵 is equal
to 𝐴𝐷𝐶 and we know that 𝐴𝐵𝐸 is equal to 𝐴𝐶𝐷.
Okay, so great, we know that our
triangles are similar. So therefore, we know that one is
going to be an enlargement of the other. So we need to first of all find out
the scale factor. Well, we know the scale factor of
any enlargement is equal to the new length divided by the original length. Well, the two corresponding lengths
we know are 6.8 and 8.5. So therefore, we can say that the
new length is gonna be 6.8 because actually we’re gonna go from the bigger triangle
to the smaller triangle. And then, that means the original
length was 8.5. So therefore, we can say that the
scale factor is equal to 6.8 divided by 8.5, which gives us a scale factor of
So that means we’re actually now at
a position to find 𝐸𝐵 because 𝐸𝐵’s corresponding side is the side 𝐷𝐶. So therefore, we can say that the
length of 𝐸𝐵 is gonna be equal to our scale factor multiplied by 𝐷𝐶, which is
gonna be equal to 0.8 multiplied by 9.5. So therefore, we can say that 𝐸𝐵
is equal to 7.6 centimeters. Okay, great, we found that let’s
move on to part b.
Now, what we need to do is actually
find the length of 𝐴𝐷. Well, to actually find this length,
the question gives us this extra bit of information. It tells us that 𝐴𝐸 is equal to
6.2 centimeters. So I’ve now marked this on the
graph. Okay, so let’s use this to actually
find out the length of 𝐴𝐷.
Well, the first thing to note is
that 𝐴𝐷 and 𝐴𝐸 are actually corresponding sides on our similar triangles. So therefore, the length of 𝐴𝐷
multiplied by a scale factor is gonna be equal to the length of 𝐴𝐸. So therefore, if we actually divide
both sides of our equation by the scale factor, we can say that 𝐴𝐷 is gonna be
equal to 𝐴𝐸 divided by our scale factor, which therefore is gonna give us that
𝐴𝐷 is equal to 6.2 divided by our scale factor, which was 0.8. So therefore, we can say that the
length of 𝐴𝐷 is equal to 7.75 centimeters. And there, we’re done. We’ve actually answered part a and