Question Video: Using the Similarity between Two Shapes to Find the Scale Factor of One Shape to the Other and the Perimeter of One of Them Mathematics • 8th Grade

Video Transcript

If 𝐴𝐵𝐶𝐷𝐸 is similar to 𝑃𝑄𝑅𝑆𝑇, find the scale factor of 𝐴𝐵𝐶𝐷𝐸 to 𝑃𝑄𝑅𝑆𝑇 and the perimeter of 𝑃𝑄𝑅𝑆𝑇.

Now I said this automatically when I read the question, but firstly just a reminder that this notation means “is similar to.” So we’re being told that these two polygons, which are pentagons, are similar to one another. This means that two things are true. Corresponding pairs of angles between the two polygons are congruent, and corresponding pairs of sides are in the same ratio. Now we do need to be a little careful here when looking at the two figures because we know that the order of letters is important when discussing similar polygons.

So if 𝐴𝐵𝐶𝐷𝐸 is similar to 𝑃𝑄𝑅𝑆𝑇, then this means that the side 𝐴𝐵, for example, corresponds to the side 𝑃𝑄. On the figure, we can see that these two sides are actually in different positions on the two polygons. In fact, they’ve been presented as reflections. So we just need to be careful when matching up pairs of corresponding sides and pairs of corresponding angles. It actually won’t really make any practical difference because each of these polygons are symmetrical. But it is important in general to make sure we’re careful when we’re matching up pairs of corresponding sides and not just to assume that the two polygons have been drawn in the same orientation.

Now we were asked to find the scale factor of 𝐴𝐵𝐶𝐷𝐸 to 𝑃𝑄𝑅𝑆𝑇. That’s moving from the larger polygon to the smaller polygon in this direction here. To calculate a scale factor, we need to divide a new length by an original length. So we’re looking to divide a length on 𝑃𝑄𝑅𝑆𝑇 by the corresponding length on 𝐴𝐵𝐶𝐷𝐸. Now on the figure, we can see that 𝑃𝑇 corresponds to 𝐴𝐸. And whilst we haven’t been given the length of 𝐴𝐸 explicitly, we can see that it is the same as the length 𝐶𝐷. So it’s also 21 units. The scale factor then, dividing a new length on 𝑃𝑄𝑅𝑆𝑇 by an original length on 𝐴𝐵𝐶𝐷𝐸, is 14 over 21. And dividing both the numerator and denominator by seven, this simplifies to two-thirds.

This value makes sense intuitively. We’re moving from the larger polygon to the smaller one. And so our scale factor should be a fractional value less than one, and two-thirds is. So we’ve answered the first part of the question. And now we need to answer the second, which is to calculate the perimeter of 𝑃𝑄𝑅𝑆𝑇. We therefore need to know all of the side lengths.

We can work out the side length of 𝑅𝑆 straight away because it’s the same as the side length 𝑃𝑇; it’s 14 units. To calculate the other side lengths, we need to use the scale factor we’ve just worked out. Side 𝑃𝑄 is corresponding with side 𝐴𝐵. So we can calculate its length by taking the length of 𝐴𝐵, 24, and multiplying by our scale factor of two-thirds. We can cancel a factor of three from the numerator and denominator, giving eight multiplied by two, which is of course equal to 16.

So we have the length 𝑃𝑄. And on the diagram, we can see that the length 𝑅𝑄 is the same as this. So it’s also 16 units. The final length we need is 𝑆𝑇, which is corresponding with the side 𝐷𝐸 on the larger polygon. Notice that it is 𝐷𝐸 and not 𝐸𝐷 because the two polygons, remember, have been drawn as reflections of one another. So to calculate 𝑆𝑇, we take the length of 𝐷𝐸, which is 27, and multiply it by our scale factor of two-thirds. Once again, we can cancel a factor of three, giving 𝑆𝑇 equals nine multiplied by two, which is 18 units.

We now have the lengths of all of the sides of 𝑃𝑄𝑅𝑆𝑇. So we’re able to calculate its perimeter. It’s equal to 16 plus 16 plus 14 plus 18 plus 14, which is 78 units. We can conclude then that for these two similar polygons, the scale factor of 𝐴𝐵𝐶𝐷𝐸 to 𝑃𝑄𝑅𝑆𝑇, so from the larger polygon to the smaller, is two-thirds. And the perimeter of 𝑃𝑄𝑅𝑆𝑇 is 78 units.