Let's go through each of the problems from the image you provided:

Problem 5

You are given two similar figures, Figure X and Figure Y, and you are asked to determine which proportion must be true.

For similar figures, the ratio of corresponding sides must be the same. The dimensions provided are:

• Figure X: 12 mm, 4.9 mm, and unknown $r$
• Figure Y: 15 mm, 5.6 mm, and 13.5 mm

Looking at the dimensions, the correct corresponding ratio would be: $\frac{5.6}{6.4} = \frac{8.4}{r}$ This leads to the correct answer, A.

Problem 6

Quadrilateral YUCK is similar to quadrilateral FISH. You're asked to find which proportion must be true.

For similar quadrilaterals, corresponding sides are proportional. The corresponding sides are matched by letters in each figure. The correct proportion will pair matching sides, so A gives the correct ratios: $\frac{FI}{YU} = \frac{UC}{IS}$

Answer: A.

Problem 7

Figure I and Figure II are similar polygons. You need to determine which proportion can be used to find the value of $m$.

The dimensions for each figure are:

• Figure I: $6.5 \, \text{in}, 4 \, \text{in}, 8 \, \text{in}, 3.1 \, \text{in}$
• Figure II: $13 \, \text{in}, 6 \, \text{in}, 8 \, \text{in}, 6.2 \, \text{in}$

The proportion that should hold is: $\frac{13}{6.5} = \frac{m}{6}$ So, the correct answer is B.

Problem 8

Triangle CAT is similar to triangle DOG, and you need to find the correct proportion.

For similar triangles, corresponding sides must be proportional, so the correct proportion is: $\frac{CA}{DO} = \frac{OG}{AT}$ Answer: A.

Would you like a detailed explanation of how the proportions were calculated?

Here are 5 related questions:

1. What properties define similar polygons?
2. How can you use cross-multiplication to solve proportions in geometry?
3. Can you explain how scaling factors work with similar figures?
4. What is the relationship between angles and side lengths in similar figures?
5. How can you determine if two triangles are similar?

Tip: Always check that corresponding sides and angles are correctly matched when solving for proportions in similar figures.