Question Video: Determining Similar Triangles by Considering Their Corresponding Sides’ Lengths

Given the following four shapes, which two are similar? [A] Shape 1 [B] Shape 2 [C] Shape 3 [D] Shape 4

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Video Transcript

Given the following four shapes, which two are similar?

To answer this, we can recall that similar shapes are the same shape but different sizes. For shapes to be similar, they must have corresponding sides which are in the same proportion. So, let’s take a closer look at our four shapes. We can see that shape one and shape three look like they are the same shape. Shapes two and four also look like these could potentially be similar.

Taking the first pair, shape one and three, let’s have a look at the sides of this shape. Looking at the base of these triangles then, labeled eight on shape one and four on shape three, then to go from shape one to shape three, we would half it. So, we could say that the proportionality would be multiplying by a half. This is the same as dividing by two.

We now need to check if this proportionality works for all sides. Taking our vertical side, we can see that if we take six and multiply by half, we would indeed get three. Taking our final length in, we’re looking at 10 and thinking if we half it, would we get five? And the answer is yes. This means that all the corresponding sides are in the same proportion. Therefore, shape one and shape three are similar.

We can now check if shape two and shape four are also similar. Starting with the base of these triangles, then to go from five to eight, we would need to multiply by eight and then divide by five, which is the same as multiplying by eight over five.

We now need to check if this proportionality works for the remaining two sides. Comparing the lengths of seven and 12, then we need to check if seven times eight over five equals 12. To calculate seven times eight over five, we can write our seven as a fraction over one and multiply the numerators and denominators, which gives 56 over five. As a mixed-number fraction, this would be equal to 11 and one-fifth. Therefore, seven times eight-fifths does not equal 12, and so our proportionality does not work. So, shape two and shape four are not similar.

Therefore, our final answer for the two similar shapes are shape one and shape three.

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