Question

the figures shown are similar. find lengths of x, y, and z.
the length of side x is 12.
(type an integer or a decimal.)
the length of side y is
(type an integer or a decimal.)
images of two similar quadrilaterals: top one has sides x, y, 8, z; bottom one has sides 24, 32, 16, 24

Explanation:

Step1: Find the scale factor

The corresponding sides of similar figures are proportional. Let's take the sides with lengths 24 (in the larger figure) and 12 (in the smaller figure, wait no, wait the larger figure has a side of 24 and the smaller figure's corresponding side x is 12? Wait, no, looking at the figures, the larger figure has a vertical side of 24, and the smaller figure's vertical side is x = 12. So the scale factor from the smaller to the larger is \( \frac{24}{12} = 2 \). Wait, or maybe the other way. Wait, the larger figure has a side of 24 (bottom base) and the smaller figure's bottom base is z? Wait, no, the problem says the length of x is 12. Let's look at the sides: the larger figure has vertical side 24, smaller has x = 12. So the ratio of larger to smaller is \( \frac{24}{12} = 2 \). Now, the larger figure has a slant side of 32, and the smaller has y. So we can set up the proportion: \( \frac{32}{y} = 2 \) (since larger is 2 times smaller). Wait, or maybe the ratio is smaller to larger. Wait, let's check another side. The larger figure has a side of 16, smaller has 8. So \( \frac{16}{8} = 2 \). So the scale factor from smaller to larger is 2. So the larger figure's sides are 2 times the smaller's. So the smaller figure's slant side y corresponds to larger's 32. So \( y \times 2 = 32 \), so \( y = \frac{32}{2} = 16 \)? Wait, no, wait the larger figure's slant side is 32, smaller is y. So if the scale factor is 2 (larger is 2x smaller), then \( y = \frac{32}{2} = 16 \)? Wait, but let's confirm with another side. The larger figure's vertical side is 24, smaller is x = 12. 24 is 2x 12, correct. Larger's side 16, smaller's 8. 16 is 2x 8, correct. Larger's bottom base 24, smaller's z. So z x 2 = 24, so z = 12. Now, the slant side of larger is 32, smaller is y. So y x 2 = 32, so y = 16. Wait, but let's do it properly. Let's denote the smaller figure's sides: x (vertical) = 12, z (bottom base), y (slant), 8 (the other side). Larger figure: 24 (vertical), 24 (bottom base), 32 (slant), 16 (the other side). So the ratio of corresponding sides: vertical: 24/12 = 2, other side: 16/8 = 2, bottom base: 24/z = 2 ⇒ z = 12, slant: 32/y = 2 ⇒ y = 16. So that's correct.

Step2: Solve for y

Using the proportion from similar figures: \( \frac{\text{Length in larger figure}}{\text{Length in smaller figure}} = \text{scale factor} \). We know from the side with length 8 (smaller) and 16 (larger) that the scale factor is \( \frac{16}{8} = 2 \). So for the slant side: larger length is 32, smaller is y. So \( \frac{32}{y} = 2 \). Solving for y: \( y = \frac{32}{2} = 16 \).

Answer:

16