Question Video: Checking Whether Two Given Triangles Are Similar or Not given Their Dimensions | Nagwa Question Video: Checking Whether Two Given Triangles Are Similar or Not given Their Dimensions | Nagwa

Question Video: Checking Whether Two Given Triangles Are Similar or Not given Their Dimensions Mathematics • Second Year of Preparatory School

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Is triangle 𝑀𝑁𝐿 similar to triangle π‘‹π‘π‘Œ?

02:03

Video Transcript

Is triangle 𝑀𝑁𝐿 similar to triangle π‘‹π‘π‘Œ?

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 π‘‹π‘Œ, which is also equal to the proportion 𝑁𝐿 over π‘π‘Œ, 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.

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