Video Transcript
Determine the length of line
segment 𝐴𝐶.
In the figure, we have two
triangles: 𝑥𝑦𝑧 on the left and 𝐴𝐵𝐶 on the right. We are asked to find the length of
line segment 𝐴𝐶, which appears in triangle 𝐴𝐵𝐶.
Now we might notice that we only
have the length of one other side in this triangle, which won’t be enough on its own
to work out line segment 𝐴𝐶. We might therefore wonder if the
two triangles are similar. We can recall that similar
triangles have corresponding angles congruent and corresponding sides in
proportion. So if the triangles are similar,
this would give us a way to work out any unknown side lengths.
If we want to prove that two
triangles are similar, we can either demonstrate that all corresponding angles are
congruent or that all pairs of corresponding sides are in proportion. If we look at the triangles, we
only have one pair of sides in the triangles given. That wouldn’t be enough to prove
they are similar. So let’s consider the angles.
We can observe that the measures of
both angles 𝑧 and 𝐶 are given as 61 degrees. We are given that the measure of
angle 𝐴 is 59 degrees, but we aren’t given the measure of angle 𝑥. However, let’s see if we can
calculate it. We know that the angle measures in
a triangle sum to 180 degrees. So the measure of angle 𝑥 will be
180 degrees subtract the sum of 61 degrees and 60 degrees, as these are the two
remaining angles in the triangle. This gives us 59 degrees. So the measure of angle 𝑥 is equal
to the measure of angle 𝐴.
Next, let’s consider the measure of
angle 𝑦, which is given as 60 degrees. Its corresponding angle in triangle
𝐴𝐵𝐶 will be angle 𝐵. And the measure of angle 𝐵 is
equal to 180 degrees subtract the sum of 59 degrees and 61 degrees, which is 60
degrees. And so the measures of angle 𝑦 and
𝐵 are equal.
We have therefore demonstrated that
we have three pairs of corresponding angles congruent, so the triangles are
similar. Writing this similarity
relationship carefully, remembering to make sure that the corresponding vertices are
given in the correct place in the similarity relationship, we can say that triangle
𝑥𝑦𝑧 is similar to triangle 𝐴𝐵𝐶. We can now use this to help us work
out the unknown side length of line segment 𝐴𝐶.
We were given the lengths of two
corresponding sides: 𝑧𝑦 and 𝐶𝐵. We could write the proportion of
the sides as 𝐶𝐵 over 𝑧𝑦. Now the side 𝑥𝑧 is corresponding
to the side 𝐴𝐶 that we need to calculate. And because the triangles are
similar, their sides are in proportion, so 𝐶𝐵 over 𝑧𝑦 equals 𝐴𝐶 over 𝑥𝑧. We can then fill in the
measurements for the lengths. So we have 22.8 over 12 equals 𝐴𝐶
over 12.1.
We can simplify the left-hand side
first to 11.4 over six and then multiply both sides by 12.1, which gives us the
answer that the length of line segment 𝐴𝐶 is 22.99 centimeters.
For a good, approximate check on
the answer, we can consider the sides. In triangle 𝑥𝑦𝑧, sides 𝑧𝑦 and
𝑥𝑧 are 12 and 12.1 centimeters, respectively, which are quite close in value, and
side 𝑥𝑧 is slightly longer. We would expect that sides 𝐶𝐵 and
𝐴𝐶 would follow the same pattern. That is, they are quite close in
value with the side 𝐴𝐶 being slightly longer. And 22.99 centimeters would seem
like an appropriately sized value. So 𝐴𝐶 is 22.99 centimeters.
