Question

the figures shown are similar. find the lengths of x, y, and z.

the length of side x is 5.25.
(type an integer or a decimal.)
the length of side y is
(type an integer or a decimal.)

Explanation:

Step1: Find the scale factor

The corresponding sides of similar figures are proportional. Let's find the scale factor using the sides of length 3 and 4 (or 7 and 7, but 3 and 4 are another pair). Wait, actually, the side of length 3 in the smaller figure corresponds to 4 in the larger? Wait no, wait the smaller figure has a side of 3 and the larger has 4? Wait no, looking at the figures: the smaller quadrilateral has a side of 3, the larger has 4. The smaller's x corresponds to larger's 7? Wait, x is 5.25, and larger's corresponding side is 7. Wait, 5.25 / 7 = 0.75. Let's check another pair: the larger has a side of 4, smaller has 3. 3/4 = 0.75? Wait 3 divided by 4 is 0.75? Wait no, 3/4 is 0.75? Wait 3 divided by 4 is 0.75? Wait 3 ÷ 4 = 0.75. Wait the larger figure has a side of 8, and the smaller's y corresponds to 8? Wait no, the larger figure's side is 8, and the smaller's y is the corresponding side. Wait the scale factor is the ratio of corresponding sides. Let's take the side of length 4 in the larger figure and 3 in the smaller: 3/4 = 0.75. Let's check x: x is 5.25, and the larger's corresponding side is 7. 5.25 / 7 = 0.75. Yes, that works. So the scale factor from larger to smaller is 0.75, or from smaller to larger is 4/3? Wait no, wait x is 5.25, which is the smaller's side, and the larger's corresponding side is 7. So 5.25 = 7 (3/4)? Wait 7 0.75 = 5.25. Yes. So the ratio of smaller to larger is 3/4 (since 3 corresponds to 4, 5.25 corresponds to 7, so 3/4 = 5.25/7 = 0.75). So now, the larger figure has a side of 8, and the smaller's y corresponds to 8. So y = 8 (3/4) = 6. Wait let's check: 8 0.75 = 6. Yes. So that's the length of y.

Step1: Determine the scale factor

The ratio of corresponding sides (smaller to larger) is $\frac{3}{4}$ (since 3 in smaller corresponds to 4 in larger, and 5.25 in smaller corresponds to 7 in larger: $\frac{5.25}{7} = 0.75 = \frac{3}{4}$).

Step2: Calculate the length of y

The larger figure has a side of length 8, and y is the corresponding side in the smaller figure. Using the scale factor $\frac{3}{4}$, we have:
$y = 8 \times \frac{3}{4} = 6$

Answer:

6