Are these two polygons similar? If yes, find the scale factor from polygon 𝑋𝑌𝑍𝐿 to polygon 𝐴𝐵𝐶𝐷.
So we’re asked to determine whether the two quadrilaterals are similar. Let’s recall the criteria that are necessary for two polygons to be similar. Firstly, it’s necessary that corresponding angles in the two polygons are congruent. Secondly, it’s necessary that corresponding side lengths are proportional. We’ll take each of these statements in turn, beginning with the angles.
We can see from the way the diagram has been marked that three of the angles in the two polygons are indeed congruent. The final angle must also be congruent, as the angle sum in a quadrilateral is always 360 degrees. Therefore, the first condition for similarity is fulfilled. Corresponding angles between the two polygons are indeed congruent.
Now let’s consider the proportionality of the corresponding side lengths. Pairs of corresponding sides have now been marked in the same color. And so we need to determine whether it’s true that the ratio between these pairs of sides is constant for all four pairs. We’ll begin with 𝐷𝐶 divided by 𝐿𝑍 which is 2.56 divided by 3.2. This simplifies to just 0.8.
What we now want to determine is whether the three remaining pairs of sides give the same ratio. The measurements for 𝐶𝐵 and 𝑍𝑌 are in fact the same as the measurements for 𝐷𝐶 and 𝐿𝑍. So we can conclude straightaway that they will give the same ratio. We need to look at the other two pairs of sides. The ratio of 𝐵𝐴 to 𝑌𝑋 is 3.84 divided by 4.8. This also simplifies to 0.8.
So far, we have three pairs of corresponding side lengths which are proportional, with a scale factor of 0.8. We need to check the final pair. 𝐴𝐷 divided by 𝑋𝐿 is 2.72 divided by 3.4. And this ratio does also simplify to 0.8. So as all four pairs of corresponding sides have the same scale factor of 0.8, we can conclude that corresponding side lengths are indeed proportional. And therefore, the second criteria for similarity is also fulfilled.
So our answer to the problem then is that yes, the two polygons are similar. And the scale factor from 𝑋𝑌𝑍𝐿, which is the larger polygon, to 𝐴𝐵𝐶𝐷, the smaller polygon, is 0.8.