If two triangles are similar and, additionally, the ratio between the lengths of two corresponding sides is one to one, what can you say about the triangles? Option A) They are congruent. Option B) They’re not congruent. Option C) They have different areas.
So before we start answering this question, let’s remind ourselves of what the word “similar” means. And “similar” means the same shape. So let’s draw two similar triangles. And since they’re similar, this means that corresponding angles will be equal. There will also be a ratio of sizes from one triangle to the other. For example, if the ratio in our drawing here was one to two, that means every corresponding side on the large triangle would be twice as big as it is on the smaller triangle.
So if we had a length of three centimeters on our smaller triangle, then the corresponding length on the larger triangle would be six centimeters. However, here, the question tells us that the ratio of sides is one to one. So our pink lengths would be the same size, our orange lengths would be the same size, and the green lengths would be the same size. And so, our diagram should look a little bit more like this. So we’re told the triangles are similar. And since a ratio is one to one, that means that the corresponding sides are exactly the same length. So we can say that the triangles are the same shape and same size. And since being the same shape and size is the definition for congruency, we can say that our triangles are congruent. So in our question then, we know that option A applies. But let’s check option B.
Well, if our triangles are congruent, then they cannot also be not congruent. So option B does not work. Let’s check option C. To find the area of a triangle, we calculate half times 𝑏 times ℎ, where 𝑏 is the base and ℎ is the height. In our triangles, we know that since they’re the same size, they both have the same base length and the same height. This means that they would have exactly the same areas. So option C would not be correct.
And so, our final answer is then if our triangles are similar with the ratio of corresponding sides one to one, we can say they are congruent.