Video Transcript
A triangle ๐ด๐ต๐ถ is rotated by 180
degrees about the origin to triangle ๐ด prime ๐ต prime ๐ถ prime. Are triangles ๐ด๐ต๐ถ and ๐ด prime
๐ต prime ๐ถ prime similar? Are triangles ๐ด๐ต๐ถ and ๐ด prime
๐ต prime ๐ถ prime congruent?
Weโre told here that we have a
triangle ๐ด๐ต๐ถ which is rotated 180 degrees to give triangle ๐ด prime ๐ต prime ๐ถ
prime. Weโre then asked two questions
about this original triangle and its image: firstly, if theyโre similar and
secondly, if theyโre congruent. Letโs begin by thinking a little
bit more about this rotation. Weโre not told anything about what
๐ด๐ต๐ถ 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 ๐ด prime ๐ต prime
๐ถ prime the image would look like this. In the second example, our image ๐ด
prime ๐ต prime ๐ถ prime would look like this. So, letโs 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
๐ด and the angle at ๐ด prime. In both of our examples, these two
angles would be the same size. Angles ๐ต and ๐ต prime are equal,
and angles ๐ถ and ๐ถ 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 ๐ด๐ต would
be the same as the length ๐ด prime ๐ต 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, ๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime would be
similar. Weโve seen that it doesnโt matter
what triangle ๐ด๐ต๐ถ looks like, what size it is, or where it is. Our rotation would create similar
shapes.
To answer the second part of this
question, letโs 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 itโs different to similar
triangles because weโre saying that the corresponding sides have to be equal. In other words, congruent triangles
are the same shape and the same size.
Weโve 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 ๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime are
congruent. Itโs worth remembering that in a
rotation, the object and its image will always be congruent.
