Question

1. on a map, the distance between two towns is 1.5 inches. however, the actual distance between the two towns is 6 miles.

a. two other towns are 18 miles apart. how far apart would they be on the map?
b. is it accurate to say that the actual distance between two locations is always 4 times greater than the distance shown on the map? why or why not?

1. are the following statements always, sometimes, or never true?

a. given any two squares with different side lengths, the squares are similar figures.
b. given any two rectangles with different side lengths, the rectangles are similar figures.

1. are the following statements always, sometimes, or never true?

a. two triangles with different side lengths are similar figures.
b. two triangles with different side lengths are congruent figures.

Explanation:

Step1: Find map scale factor

First, calculate the map scale: $\frac{\text{Actual Distance}}{\text{Map Distance}} = \frac{6 \text{ miles}}{1.5 \text{ inches}} = 4 \text{ miles per inch}$

Step2: Solve 5a: Find map distance

Use scale to find map distance: $\text{Map Distance} = \frac{\text{Actual Distance}}{\text{Scale}} = \frac{18 \text{ miles}}{4 \text{ miles per inch}} = 4.5 \text{ inches}$

Step3: Answer 5b: Verify scale claim

Check the ratio: $\frac{\text{Actual Distance}}{\text{Map Distance}} = 4$, so actual distance is 4× map distance.

Step4: Answer 6a: Square similarity rule

All squares have 4 right angles, sides are proportional.

Step5: Answer 6b: Rectangle similarity rule

Rectangles only have right angles; side ratios may not match.

Step6: Answer 7a: Triangle similarity rule

Triangles need proportional sides/equal angles to be similar.

Step7: Answer 7b: Triangle congruence rule

Congruent triangles must have identical side lengths.

Answer:

5a. 4.5 inches
5b. Yes, it is accurate. The ratio of actual distance to map distance is a constant $\frac{6}{1.5}=4$, so actual distance is always 4 times the map distance.
6a. Always true
6b. Sometimes true
7a. Sometimes true
7b. Never true