Video Transcript
𝐴𝐵𝐶𝐷 is a rectangle in which
𝐴𝐷 equals 21 centimeters, 𝐴𝑋 equals nine centimeters, and 𝑋𝑀 equals 12
centimeters. Calculate the perimeter of triangle
𝑌𝑀𝐶.
Let’s begin this question by
filling in the length information that we are given onto the diagram. We have 𝐴𝐷 as 21 centimeters,
𝐴𝑋 is nine centimeters, and 𝑋𝑀 is 12 centimeters. We need to calculate the perimeter
of triangle 𝑌𝑀𝐶, which is in the lower part of the diagram.
We are given the information that
𝐴𝐵𝐶𝐷 is a rectangle. So opposite sides are parallel and
congruent. Therefore, the line segments 𝐴𝐷
and 𝐵𝐶 are parallel and line segments 𝐴𝐵 and 𝐶𝐷 are parallel. We can also observe from the
markings on the diagram that line segment 𝑋𝑌 is also parallel to line segment
𝐴𝐵. So, in fact, there are three
parallel line segments here.
Now, it might be useful to consider
if we have similar triangles in this figure. Specifically, we might pose the
question “Is triangle 𝑋𝑀𝐴 similar to triangle 𝑌𝑀𝐶?” We can recall that similar
triangles have corresponding angles congruent and corresponding sides in
proportion. One way we can prove that two
triangles are similar is by demonstrating that all corresponding pairs of angles are
congruent. So let’s see what we can
determine.
Using the parallel line segments
𝐴𝐷 and 𝐵𝐶 and the transversal 𝐴𝐶, we can note that angles 𝑋𝐴𝑀 and 𝑌𝐶𝑀
are alternate angles and are therefore equal in measure. Next, we have that angles 𝑋𝑀𝐴
and 𝑌𝑀𝐶 are vertically opposite angles. So these will also be
congruent. So what about the third pair of
angles in each triangle?
Well, since 𝐴𝐵𝐶𝐷 is a
rectangle, we know that it has four angles with a measure of 90 degrees. And because we have the three
parallel line segments of 𝐴𝐵, 𝐶𝐷, and 𝑋𝑌 and the transversals 𝐴𝐷 and 𝐵𝐶,
then we know that the measures of angles 𝐴𝑋𝑀 and 𝐶𝑌𝑀 will both be 90
degrees. So having these three pairs of
angles congruent proves that triangles 𝑋𝑀𝐴 and 𝑌𝑀𝐶 are similar.
We can now use this information to
help us work out the perimeter of triangle 𝑌𝑀𝐶. One way to do this is to use the
fact that the ratio between the perimeters of two similar triangles is equal to the
ratio between any two corresponding sides, which is the similarity ratio. In other words, if we have two
similar triangles and we worked out the scale factor or similarity ratio between
pairs of corresponding sides of triangles one and two and called that 𝑎, then the
perimeters would also have the same similarity ratio of 𝑎.
So one way we could calculate the
perimeter of triangle 𝑌𝑀𝐶 is by first calculating the perimeter of triangle
𝑋𝑀𝐴 and multiplying it by the similarity ratio between these two triangles. If we consider triangle 𝑋𝑀𝐴, we
notice that we have two sides whose length we know. But as this is a right triangle, we
can find the third length by applying the Pythagorean theorem.
The Pythagorean theorem states that
in a right triangle, the square of the hypotenuse is equal to the sum of the squares
of the other two sides. The two shorter sides in triangle
𝑋𝑀𝐴 are 𝐴𝑋 and 𝑋𝑀, and the hypotenuse is 𝐴𝑀, which is the side we need to
calculate. We can fill in the side lengths of
nine and 12 centimeters and evaluate the squares. Simplifying, we have that 225
equals 𝐴𝑀 squared. We can then take the square root of
both sides. And since 𝐴𝑀 is a length, we know
that it must be a positive value. So 𝐴𝑀 is 15 centimeters. As the perimeter is the distance
around the outside edge, we can find the perimeter of triangle 𝑋𝑀𝐴 by adding the
three lengths of nine, 12, and 15 centimeters, which gives 36 centimeters.
So, now, if we return to our
calculation for the perimeter of triangle 𝑌𝑀𝐶, we note that we still need to find
the similarity ratio between triangles 𝑋𝑀𝐴 and 𝑌𝑀𝐶. We can do this by identifying the
lengths of a pair of corresponding sides in these triangles. It does appear as though we don’t
know any side lengths in triangle 𝑌𝑀𝐶. However, we can return to the fact
that 𝐴𝐵𝐶𝐷 is a rectangle. So line segment 𝐵𝐶 is also 21
centimeters, since it is opposite line segment 𝐴𝐷. And since 𝐴𝑋𝑌𝐵 is also a
rectangle, then the opposite sides 𝐴𝑋 and 𝐵𝑌 are both nine centimeters. The length of line segment 𝐶𝑌 can
then be calculated as 21 minus nine centimeters, which is 12 centimeters.
In our similar triangles, the sides
𝐶𝑌 and 𝐴𝑋 are corresponding. So, to find the similarity ratio
from triangle 𝑋𝑀𝐴 to triangle 𝑌𝑀𝐶, we write the side in triangle 𝑌𝑀𝐶, which
is 𝐶𝑌, over the side in triangle 𝑋𝑀𝐴, which is 𝐴𝑋. Filling in the side lengths, we get
the ratio 12 over nine, which can be simplified to four-thirds.
This similarity ratio means that
every side length in triangle 𝑌𝑀𝐶 is four-thirds the corresponding side length in
triangle 𝑋𝑀𝐴. And as we previously mentioned, the
perimeter of triangle 𝑌𝑀𝐶 is four-thirds the perimeter of triangle 𝑋𝑀𝐴. The perimeter of triangle 𝑋𝑀𝐴 is
36 centimeters. So the perimeter of triangle 𝑌𝑀𝐶
is 48 centimeters. And so we have found the answer by
first proving that the triangles are similar.
As an aside, an alternative method
would be to determine the side lengths of triangle 𝑌𝑀𝐶 by using the similarity
ratio of four-thirds and then adding them to find the perimeter. Sides 𝑋𝑀 and 𝑌𝑀 are
corresponding. So four-thirds times 12 centimeters
gives 16 centimeters. Sides 𝐴𝑀 and 𝐶𝑀 are
corresponding. So four-thirds times 15 centimeters
gives 20 centimeters. And adding the three lengths of
triangle 𝑌𝑀𝐶, which are 12, 16, and 20 centimeters, would also have directly
given us the answer of 48 centimeters.
