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 6.
(type an integer or a decimal.)
the length of side z is
(type an integer or a decimal.)

Explanation:

Step1: Determine the scale factor

The corresponding sides of similar figures are proportional. Let's find the scale factor using the sides of length 4 and 3 (or other corresponding sides). The scale factor from the larger figure to the smaller figure is $\frac{3}{4}$ (since 3 corresponds to 4). We can also check with other sides: for example, the side of length 7 in the larger figure corresponds to x = 5.25 in the smaller figure. Let's verify the scale factor: $\frac{5.25}{7} = 0.75$, and $\frac{3}{4} = 0.75$, so the scale factor is 0.75 (or $\frac{3}{4}$).

Step2: Find the length of z

The side of length 7 in the larger figure corresponds to z in the smaller figure. Using the scale factor, we have $z = 7 \times 0.75$.
Calculating that: $7 \times 0.75 = 5.25$? Wait, no, wait. Wait, maybe I mixed up the correspondence. Wait, let's check the sides again. Wait, the larger figure has a side of length 4, and the smaller has 3. So the ratio of smaller to larger is $\frac{3}{4}$. The larger figure has a side of length 7 (the bottom side), so the smaller figure's bottom side z should be $7 \times \frac{3}{4}$? Wait, no, wait, let's check the other sides. Wait, the larger figure's vertical side is 7, and the smaller's vertical side is x = 5.25. So $\frac{5.25}{7} = 0.75$, which is $\frac{3}{4}$. The larger figure's top side is 8, and the smaller's top side is y = 6. $\frac{6}{8} = 0.75$, which is $\frac{3}{4}$. The larger figure's right side is 4, and the smaller's right side is 3. $\frac{3}{4} = 0.75$. So the ratio of smaller to larger is $\frac{3}{4}$. Therefore, the larger figure's bottom side is 7, so the smaller figure's bottom side z is $7 \times \frac{3}{4}$? Wait, no, wait, that would be 5.25, but let's check. Wait, maybe the larger figure's bottom side is 7, and the smaller's bottom side is z. So if the ratio is smaller/larger = 3/4, then z = 7 * (3/4)? Wait, no, wait, 7 * 0.75 is 5.25? But let's check with the other sides. Wait, the larger figure's bottom side is 7, and the smaller's bottom side is z. Let's confirm the correspondence. Let's list the corresponding sides:

- Larger vertical side: 7, Smaller vertical side: x = 5.25 (ratio 5.25/7 = 0.75)
- Larger top side: 8, Smaller top side: y = 6 (ratio 6/8 = 0.75)
- Larger right side: 4, Smaller right side: 3 (ratio 3/4 = 0.75)
- Larger bottom side: 7, Smaller bottom side: z (ratio z/7 = 0.75)

Therefore, z = 7 * 0.75 = 5.25? Wait, but that seems the same as x, but maybe that's correct. Wait, but let's check again. Wait, maybe I made a mistake in the correspondence. Wait, maybe the larger figure's bottom side is 7, and the smaller's bottom side is z. So z = 7 * (3/4) = 5.25? Wait, but let's check with the other sides. Wait, the larger figure has sides 7 (vertical), 8 (top), 4 (right), 7 (bottom). The smaller figure has sides x (vertical) = 5.25, y (top) = 6, 3 (right), z (bottom). So the correspondence is:

Vertical: 7 (larger) <-> 5.25 (smaller)

Top: 8 (larger) <-> 6 (smaller)

Right: 4 (larger) <-> 3 (smaller)

Bottom: 7 (larger) <-> z (smaller)

So the ratio is 0.75 for each. Therefore, z = 7 * 0.75 = 5.25? Wait, but that seems the same as x, but maybe that's correct. Wait, but let's calculate 7 * 0.75: 7 * 3/4 = 21/4 = 5.25. Wait, but let's check with the other ratio. Wait, maybe the ratio is larger to smaller? No, because 5.25 is smaller than 7, 6 is smaller than 8, 3 is smaller than 4. So the ratio is smaller/larger = 3/4. Therefore, z = 7 * (3/4) = 5.25. Wait, but that seems correct. Wait, but let's check again. Wait, maybe I mixed up the bottom side. Wait, the larger figure's bottom side is 7, and the smaller's bottom side is z. So if the ratio is 3/4, then z = 7 * 3/4 = 5.25. Yes, that makes sense.

Answer:

5.25