A farmer is planning to fence in a larger plot of land for her cows and a smaller plot of land for her goats.

Question

Larger rectangle labeled 24 feet by 32 feet and smaller rectangle labeled 18 feet by 24 feet

Are these similar shapes? Explain. Enter the answer.

Answer 1

To determine if the two rectangles are similar, we need to compare their side lengths to see if the ratios of corresponding sides are equal.

For the larger rectangle:

• Length: 32 feet
• Width: 24 feet

For the smaller rectangle:

• Length: 24 feet
• Width: 18 feet

Now, let's find the ratios of the corresponding sides:

1. Ratio of the lengths: \( \frac{32}{24} = \frac{4}{3} \)

2. Ratio of the widths: \( \frac{24}{18} = \frac{4}{3} \)

Since both ratios are equal (\( \frac{4}{3} \)), the two rectangles are similar.

Answer: Yes, the rectangles are similar because the ratios of corresponding side lengths are equal (\(\frac{4}{3}\)).

Answer 2

A street light pole casts a 14-foot shadow. At the same time, Ms. Diaz casts a 7-foot shadow.

Street Light with height 10 feet casting 14-foot shadow and person with unknown height casting 7-foot shadow

If the light pole is 10 feet tall, what is Ms. Diaz’s height? Enter the answer.

Answer 3

To determine Ms. Diaz's height using the information about the street light and the lengths of the shadows, we can set up a proportion based on similar triangles.

The ratios of the heights of the objects to the lengths of their shadows should be equal:

\[ \frac{\text{Height of the street light}}{\text{Length of the street light's shadow}} = \frac{\text{Height of Ms. Diaz}}{\text{Length of Ms. Diaz's shadow}} \]

Given:

• Height of the street light = 10 feet
• Length of the street light's shadow = 14 feet
• Length of Ms. Diaz's shadow = 7 feet
• Height of Ms. Diaz = \( h \) (unknown)

Setting up the proportion:

\[ \frac{10}{14} = \frac{h}{7} \]

Now, we can cross-multiply to solve for \( h \):

\[ 10 \cdot 7 = 14 \cdot h \]

\[ 70 = 14h \]

Now, divide both sides by 14:

\[ h = \frac{70}{14} = 5 \]

Thus, Ms. Diaz's height is 5 feet.

Answer: Ms. Diaz's height is 5 feet.