Question Video: Finding an Unknown Using the Similarity between Polygons Mathematics

Video Transcript

Given that the two polygons are similar, find the value of 𝑥.

When dealing with any question involving similar polygons, we know that the corresponding angles are congruent or the same and the corresponding sides are proportional. One pair of corresponding sides are 𝐽𝑊 and 𝑅𝑆. A second pair of corresponding sides are 𝐶𝐽 and 𝑃𝑅. As the corresponding sides are proportional, we know that their ratios are the same. Two 𝑥 plus six over seven 𝑥 minus seven must be equal to 24 over 28.

In order to find the value of 𝑥, we could cross multiply at this stage. However, it is easier to simplify our fractions first. We can factor the numerator and denominator of the left-hand fraction. Two 𝑥 plus six becomes two multiplied by 𝑥 plus three. And seven 𝑥 minus seven becomes seven multiplied by 𝑥 minus one. Dividing the numerator and denominator of our right-hand fraction by four gives us six over seven as 24 divided by four is six and 28 divided by four is equal to seven.

The denominators on both sides are divisible by seven. And the numerators are divisible by two. This leaves us with a simplified equation of 𝑥 plus three over 𝑥 minus one is equal to three. We can multiply both sides of this equation by 𝑥 minus one. Distributing the parentheses gives us 𝑥 plus three is equal to three 𝑥 minus three. We can then subtract 𝑥 and add three to both sides of the equation. This gives us six is equal to two 𝑥. Finally, dividing both sides by two gives us 𝑥 is equal to three.

If the two polygons are similar, the value of 𝑥 is three. We can check this by substituting three back into the expressions for the smaller shape. Two multiplied by three is equal to six, and adding six to this gives us 12. Seven multiplied by three is equal to 21 and subtracting seven gives us 14. It is therefore clear that our two polygons are similar with a scale factor of two as 12 multiplied by two is 24 and 14 multiplied by two is 28.