Question Video: Determining If a Triangle Is Similar and Congruent after a Rotation | Nagwa Question Video: Determining If a Triangle Is Similar and Congruent after a Rotation | Nagwa

Question Video: Determining If a Triangle Is Similar and Congruent after a Rotation

A triangle ๐ด๐ต๐ถ is rotated by 180ยฐ about the origin to triangle ๐ดโฒ๐ตโฒ๐ถโฒ. Are triangles ๐ด๐ต๐ถ and ๐ดโฒ๐ตโฒ๐ถโฒ similar? Are triangles ๐ด๐ต๐ถ and ๐ดโฒ๐ตโฒ๐ถโฒ congruent?

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Video Transcript

A triangle ๐ด๐ต๐ถ is rotated by 180 degrees about the origin to triangle ๐ด prime ๐ต prime ๐ถ prime. Are triangles ๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime similar? Are triangles ๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime congruent?

Weโre told here that we have a triangle ๐ด๐ต๐ถ which is rotated 180 degrees to give triangle ๐ด prime ๐ต prime ๐ถ prime. Weโre then asked two questions about this original triangle and its image: firstly, if theyโre similar and secondly, if theyโre congruent. Letโs begin by thinking a little bit more about this rotation. Weโre not told anything about what ๐ด๐ต๐ถ looks like or where it is. So, it could look like this or even like this. But what happens when we rotate it 180 degrees about the origin?

Recalling that the origin is the coordinate zero, zero, then in our first test case, the triangle ๐ด prime ๐ต prime ๐ถ prime the image would look like this. In the second example, our image ๐ด prime ๐ต prime ๐ถ prime would look like this. So, letโs look at our first question and recall what it means to have similar triangles.

In similar triangles, we say that all corresponding pairs of angles are equal and all corresponding pairs of sides are in proportion. Informally, we can think of similar triangles as being the same shape. We can take a look at the angle at ๐ด and the angle at ๐ด prime. In both of our examples, these two angles would be the same size. Angles ๐ต and ๐ต prime are equal, and angles ๐ถ and ๐ถ prime are equal. That means that all the pairs of corresponding angles are equal. But what about the sides?

Well, as we have simply rotated it, the lengths will be the same. For example, the length ๐ด๐ต would be the same as the length ๐ด prime ๐ต prime. We could see that all the corresponding sides would be in proportion. The proportion or scale factor here would be one. And so, our answer to the first part of this question is yes, ๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime would be similar. Weโve seen that it doesnโt matter what triangle ๐ด๐ต๐ถ looks like, what size it is, or where it is. Our rotation would create similar shapes.

To answer the second part of this question, letโs remind ourselves about congruent triangles. We say that two shapes or triangles are congruent if all pairs of corresponding angles are equal and all pairs of corresponding sides are equal. Note that itโs different to similar triangles because weโre saying that the corresponding sides have to be equal. In other words, congruent triangles are the same shape and the same size.

Weโve already demonstrated, as we had in similar triangles, that all our corresponding angles are equal. We also saw that all our corresponding sides or lengths would be equal. The answer to the second part of this question is therefore yes, triangles ๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime are congruent. Itโs worth remembering that in a rotation, the object and its image will always be congruent.

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