Question Video: Finding the Scale Factor between Two Polygons Given Their Dimensions Mathematics

Video Transcript

If 𝐴𝐵𝐶𝐷 is similar to 𝐸𝐹𝐺𝐻, find the scale factor from 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻 and the values of 𝑋 and 𝑌.

In this problem, we are told that the two given quadrilaterals, 𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐺𝐻, are similar. We can recall that two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion. We can recognize that similar polygons can be considered as a dilation of each other. If the scale factor is one, the polygons could instead be called congruent.

Here, the first part of the question asks us to find the scale factor of similarity. This is equivalent to finding the proportion that corresponding sides are in. So, let’s see if we can identify a corresponding pair of sides in the diagrams for which we know both their lengths.

We have side 𝐵𝐶 given as 47 centimeters and side 𝐹𝐺 given as 18.8 centimeters. Now, because we are told to find the scale factor from 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻, then the way in which we write the proportion as a fraction is very important. Because we are going to 𝐸𝐹𝐺𝐻, then the side 𝐹𝐺 from this quadrilateral is on the numerator. And because we are coming from 𝐴𝐵𝐶𝐷, then side 𝐵𝐶 is written on the denominator. Once we have that, then we can fill in the respective lengths of 𝐹𝐺 and 𝐵𝐶 as 18.8 and 47 centimeters.

Next, we need to simplify this fraction. If we are doing this without the use of a calculator, then often it is easiest to get rid of the decimal value of 18.8. So, multiplying both the numerator and denominator by 10 would give us 188 over 470. We might then choose to halve each of the numerator and denominator to give 94 over 235 and then divide each of these by 47. Or alternatively, we may have divided 188 and 470 by 94 and arrived at this same fully simplified value of two-fifths.

By using either method, we have found that the proportion of the corresponding sides is two-fifths. And it is also the scale factor from 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻. That means if we took any of the side lengths in 𝐴𝐵𝐶𝐷 and multiplied it by two-fifths, we would get the corresponding side length in 𝐸𝐹𝐺𝐻. Knowing this value will now allow us to determine the values of 𝑋 and 𝑌.

Let’s identify that 𝑋 is the length of the side 𝐺𝐻. And the corresponding side in 𝐴𝐵𝐶𝐷 is 𝐶𝐷. And so, if we travel from 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻, we’ll be multiplying 34 centimeters by two-fifths to give 𝑋 centimeters. As a fraction, we can calculate the left-hand side of 34 times two-fifths as 68 over five. And writing this as a mixed number fraction, we have 13 and three-fifths equals 𝑋. As a decimal then, we have 13.6 equals 𝑋. So, we have determined that the value of 𝑋 is 13.6.

Next, let’s calculate the value of 𝑌. 𝑌 can be found as part of the length of 𝐴𝐵, and the corresponding side on 𝐸𝐹𝐺𝐻 is the side 𝐸𝐹. Now, we could determine the length of 𝐴𝐵 using the scale factor of two-fifths, or we could find the scale factor in the opposite direction. That is, what is the scale factor from 𝐸𝐹𝐺𝐻 to 𝐴𝐵𝐶𝐷? Well, if we have a scale factor in one direction and we want to find the scale factor in the opposite direction, we calculate its reciprocal. The reciprocal of two-fifths is five over two. So, the scale factor from 𝐸𝐹𝐺𝐻 to 𝐴𝐵𝐶𝐷 is five over two.

So, we can write that the side 𝐸𝐹, which is 19.2, multiplied by five over two equals side 𝐴𝐵, which is 𝑌 plus four. We can then simplify the left-hand side as 96 over two. Then, we have the equation 48 is equal to 𝑌 plus four. Finally, subtracting four from both sides, we can calculate that 𝑌 is equal to 44, which is the final part of the question answered.

We can give the answers to all three parts of the question as the scale factor equals two-fifths, 𝑋 equals 13.6, and 𝑌 equals 44.