Video Transcript
Is triangle 𝑀𝑁𝐿 similar to
triangle 𝑋𝑍𝑌?
In this question, we are given two
triangles, and we need to determine if the triangles are similar. We can recall that similar
triangles have corresponding angles congruent and corresponding sides in
proportion. If we want to prove that two
triangles are similar, we can either prove that corresponding angles are congruent
or that corresponding sides are proportional. This is because having either of
these properties means that the other property is also true. So we don’t need to prove both.
Now, given that we have the lengths
of sides in the figure, let’s find the ratio between each of the sides. In each triangle, there are three
congruent side lengths, which means that they are both equilateral triangles. So if we take a pair of sides, 𝑀𝑁
and 𝑋𝑍, and write the proportion as 𝑀𝑁 over 𝑋𝑍, then this is equal to the
proportion 𝑀𝐿 over 𝑋𝑌, which is also equal to the proportion 𝑁𝐿 over 𝑍𝑌,
because we know that the proportions are all equal to the ratio 12 over 18, or
two-thirds. Therefore, all the corresponding
sides are in proportion, which means that the triangles are similar. And so we can give the answer as
yes.
Alternatively, since the triangles
are equilateral, then we know that all the angles have a measure of 60 degrees. All the corresponding angles would
be congruent. And so this property alone would
also be sufficient to prove that the triangles are similar.
