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Video: Determining Whether Two Shapes are Similar

Lauren McNaughten

The figure shows two triangles: 𝐴𝐵𝐶 and 𝐴′𝐵′𝐶′. Describe the single transformation that would map 𝐴𝐵𝐶 to 𝐴′𝐵′𝐶′. Hence, determine whether triangles 𝐴𝐵𝐶 and 𝐴′𝐵′𝐶′ are similar.

03:13

Video Transcript

The figure shows two triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime. Firstly, describe the single transformation that would map 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime. Secondly, hence, determine whether triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are similar.

Firstly, we’re asked for the single transformation that maps 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime. This word single is really important. We can’t give a combination of transformations as our answer because the question tells us that it’s possible to achieve this with just one single transformation.

Let’s look at the triangles more closely. The two triangles have different sizes, which tells us that the type of transformation we’re looking at must be a dilation. We can determine the scale factor of this dilation by looking at the length of corresponding sides in the two triangles.

The base of triangle 𝐴𝐵𝐶 is one unit, whereas the base of triangle 𝐴 prime 𝐵 prime 𝐶 prime is two units. The perpendicular height of 𝐴𝐵𝐶 is two, and the perpendicular height of 𝐴 prime 𝐵 prime 𝐶 prime is four. Therefore, the sides in triangle 𝐴 prime 𝐵 prime 𝐶 prime are always twice as long as the corresponding sides in triangle 𝐴𝐵𝐶. So the scale factor of the dilation is two.

Finally, let’s determine the coordinates of the point that this dilation has occurred from. To do this, we need to draw in lines or rays connecting the corresponding vertices of the two triangles. First we draw in the line connecting 𝐴 and 𝐴 prime. Next I draw in the line connecting 𝐶 and 𝐶 prime. Now actually, it would be enough just to draw these two rays, but I’ll draw the third in as well.

What you’ll notice is that these three rays all intersect at a common point, and it is this point that the dilation has occurred from. This point has the coordinates negative three, zero. Therefore, our answer to the first part of the question, the single transformation that maps 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime is the dilation from the point negative three, zero, with scale factor two.

The second part of the question asked us to determine whether the two triangles are similar. Well, if one triangle is an enlargement of the other, as is the case with the dilation, then all the corresponding lengths will be in the same ratio and all the corresponding angles will be the same size. So yes, the two triangles will be similar to each other.

So we have our answer to the first part of the problem; the single transformation is a dilation from the point negative three, zero, with scale factor two. And in an answer to the second part of the problem, yes the two triangles are similar.