If 𝐴𝐵𝐶𝐷 is similar to 𝐸𝐹𝐺𝐻, find the scale factor of similarity of 𝐸𝐹𝐺𝐻 to 𝐴𝐵𝐶𝐷 and the values of 𝑋 and 𝑌.
In our image, we have two quadrilaterals, one 𝐴𝐵𝐶𝐷 and another 𝐸𝐹𝐺𝐻. These two quadrilaterals are similar. In similar polygons, corresponding angles are congruent and corresponding sides are proportional. Notice here angle 𝐷 and angle 𝐻 are congruent as are 𝐶 and 𝐺. And because both of these shapes are quadrilaterals, we know that their interior angles sum to 360 degrees. Each of them have three congruent angles, which means their fourth angle will also be congruent.
This confirms that vertices 𝐴, 𝐵, 𝐶, and 𝐷 correspond to the vertices 𝐸, 𝐹, 𝐺, and 𝐻, respectively, as given in the question. We know then that side length 𝐴𝐵 corresponds to side length 𝐸𝐹. Side length 𝐵𝐶 corresponds to side length 𝐹𝐺. Side length 𝐶𝐷 corresponds to side length 𝐺𝐻. And side length 𝐷𝐴 corresponds to side length 𝐻𝐸.
If two shapes are similar, it means that their corresponding sides are all in the same proportion, and we can express this using similarity ratios. For example, the ratio of corresponding sides between the shapes are equal. The ratio of side length 𝐴𝐵 over 𝐸𝐹 will be equal to 𝐵𝐶 over 𝐹𝐺, which is also equal to 𝐶𝐷 over 𝐺𝐻 and 𝐷𝐴 over 𝐻𝐸. We can also express the corresponding relationships of side lengths within each quadrilateral. So, the ratio of side length 𝐴𝐵 over 𝐷𝐶 would be equal to 𝐸𝐹 over 𝐻𝐺. For similar polygons, we have these similarity ratios.
We could take these similarity ratios and find the values of 𝑋 and 𝑌. However, we’re being asked to find something called the scale factor of similarity. We use scale factors to multiply lengths in one of the shapes to work out corresponding lengths in the other shape. But we need to be a bit careful about how we define and describe our scale factors to make sure we are clear about our start and end points for the calculations. It works like this. If we take a side length from the polygon 𝐴𝐵𝐶𝐷 and multiply it by the scale factor, we’ll find the corresponding side lengths in 𝐸𝐹𝐺𝐻. In this case, we’re going from a larger shape to a smaller shape, in which case we would expect the scale factor to be less than one.
On the other hand, there’s a different scale factor if we’re going to take a side length from the polygon 𝐸𝐹𝐺𝐻 and to find the corresponding side length in 𝐴𝐵𝐶𝐷. In this case, we’re taking a side length from the smaller polygon and trying to find the corresponding side length in the larger polygon. And we expect that the scale factor will be greater than one. Our job here in deciding the scale factor of similarity is deciding are we trying to work from the larger to the smaller or from the smaller to the larger. At first glance of 𝐸𝐹𝐺𝐻 to 𝐴𝐵𝐶𝐷 seems to imply we’re taking the smaller polygon 𝐸𝐹𝐺𝐻 and trying to find the larger 𝐴𝐵𝐶𝐷.
Upon closer inspection, the similarity of 𝐸𝐹𝐺𝐻 to 𝐴𝐵𝐶𝐷 implies that the shape 𝐴𝐵𝐶𝐷 is the original shape and 𝐸𝐹𝐺𝐻 is the shape that is similar to it. The scale factor for this similarity is the multiplier that we should apply to the lengths in 𝐴𝐵𝐶𝐷 to find their corresponding lengths in 𝐸𝐹𝐺𝐻. That was the first option we considered, where we expected to have a scale factor of less than one. Since we have the values of two corresponding side lengths between 𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐺𝐻, we can calculate the scale factor. If we take 𝐵𝐶 from triangle 𝐴𝐵𝐶𝐷 and the corresponding side 𝐺𝐹 in 𝐸𝐹𝐺𝐻, and we let 𝑠 be our unknown scale factor, 𝐵𝐶 equals 47. 𝐹𝐺 equals 18.8.
If we divide both sides of the equation by 47, we get a scale factor of 0.4. It’s more common to write scale factors in fraction form, so we have a scale factor of two-fifths. Let’s use this information to find 𝑌 and 𝑋. 𝐴𝐵 corresponds to 𝐸𝐹, and therefore 𝑌 plus four times two-fifths equals 19.2. Dividing both sides by two-fifths, we find 𝑌 plus four equals 48. Subtracting four from both sides and we find 𝑌 equals 44.
To find the value of 𝑋, we have the corresponding side 𝐷𝐶 multiplied by the scale factor of two-fifths equals 𝐻𝐺. 34 times two-fifths equals 𝑋, which makes 𝑋 13.6. The final answer then, scale factor equals two-fifths, 𝑋 equals 13.6, and 𝑌 equals 44.