If 𝐸𝐶 equals 19 centimeters,
determine the length of 𝐸𝐷 rounded to the nearest hundredth, if necessary.
We can fill in the information on
the diagram that 𝐸𝐶 equals 19 centimeters. We can also establish that the
length 𝐸𝐷 that we wish to find out is on the top part of this diagram. It might not look particularly
obvious how we would find this length 𝐸𝐷, but there is one way we could check. If our smaller triangle 𝐸𝐶𝐴 is
similar to the larger triangle of 𝐸𝐷𝐵, then that would give us a way to find the
missing length of 𝐸𝐷.
Let’s have a look at both of these
triangles and jot down anything that we know about the angles. From the markings on the diagram,
we can see that angle 𝐴𝐶𝐸 in triangle 𝐸𝐶𝐴 is equal to angle 𝐵𝐷𝐸 in triangle
𝐸𝐷𝐵. We’re not given any information
about the angle 𝐶𝐴𝐸. So, let’s look at the other angle,
which is angle 𝐶𝐸𝐴. There is, in fact, another equal
angle to this in triangle 𝐸𝐷𝐵. And it’s this one, angle
𝐷𝐸𝐵. We know this because these two
angles are vertically opposite.
What we’ve found now then is that
there are two pairs of corresponding angles congruent in our two triangles. We can then write that triangle
𝐸𝐶𝐴 and triangle 𝐸𝐷𝐵 are similar using the 𝐴𝐴 similarity rule, where the
𝐴𝐴 stands for two pairs of corresponding angles congruent.
So, how does this help us find our
missing length of 𝐸𝐷? Well, in similar triangles, the
corresponding sides are in proportion. So, we need to find a pair of
corresponding sides and work out the proportion. We’re given the length of a pair of
corresponding sides, the length 𝐷𝐵 and the length 𝐴𝐶. So, if we were going from the
smaller triangle 𝐸𝐶𝐴 to the larger triangle 𝐸𝐷𝐵, we could find the proportion
of sides or the scale factor by working out the new length divided by the original
So, our scale factor would be our
new length, in this case, the length of 𝐵𝐷, over the length of 𝐴𝐶, which is
nine. We could simplify this fraction to
give us a scale factor of two, which means that every length on the smaller triangle
is multiplied by two to give the corresponding length on the larger triangle. We now need to work out which
length corresponds to the length of 𝐸𝐷 that we wish to find out. And it’s this length, 𝐶𝐸.
To find the length of 𝐸𝐷 then, we
take the corresponding length on triangle 𝐸𝐶𝐴, which is 𝐶𝐴. And it’s 19 centimeters. And we multiply it by the scale
factor of two. And 19 multiplied by two will give
us 38 because 10 times two is 20 and nine times two is 18. Adding those would give us 38. The length units here will still be
in centimeters. So, we found the length of 𝐸𝐷 as
38 centimeters. And we did this firstly by proving
that the two triangles were similar.