Question

practice 3.1 scale factor and proportionality of similar figures

1. a model of a skyscraper uses a scale of 1 cm = 5 meters. if the model is 60 cm tall, what is the height of the skyscraper?
2. for the triangle shown: if the scale factor were 1/2, what would be the new side lengths of the scaled - down triangle? (draw a triangle if needed)

use arrows a and b for problems 3 - 5

1. what is the scale factor from a to b?
2. explain how you know:
3. label each missing side so the lengths of arrow b are proportional to the side lengths of arrow a.

Explanation:

Step1: Solve problem 1

Set up proportion. Given scale 1 cm = 5 meters and model is 60 cm tall. Let the actual - height be $x$ meters. The proportion is $\frac{1}{5}=\frac{60}{x}$. Cross - multiply: $1\times x = 5\times60$.
$x = 300$ meters

Step2: Solve problem 2

If the scale factor is $\frac{1}{2}$, and the original side lengths of the triangle are 5, 12, and 13. Multiply each side length by the scale factor.
New side lengths: $5\times\frac{1}{2}=\frac{5}{2}$, $12\times\frac{1}{2} = 6$, $13\times\frac{1}{2}=\frac{13}{2}$

Step3: Solve problem 3

To find the scale factor from A to B, compare corresponding side lengths. Let's assume we compare the side lengths where we know the values. If we consider the side lengths of 2 in A and 4 in B, the scale factor $k=\frac{4}{2}=2$

Step4: Solve problem 4

We know the scale factor is 2 because when we compare corresponding side lengths of similar figures (Arrow A and Arrow B), the ratio of the side lengths of Arrow B to Arrow A is constant. For example, if a side of Arrow A is $a$ and the corresponding side of Arrow B is $b$, $\frac{b}{a}=2$ for all corresponding sides.

Step5: Solve problem 5

If the scale factor from A to B is 2:

• The side corresponding to the side of length 1 in A will have length $1\times2 = 2$ in B.
• The side corresponding to the side of length 2 in A will have length $2\times2 = 4$ in B.
• The side corresponding to the side of length 6 in A will have length $6\times2 = 12$ in B.

Answer:

1. The actual height of the skyscraper is 300 meters.
2. The new side lengths of the triangle are $\frac{5}{2}$, 6, and $\frac{13}{2}$.
3. The scale factor from A to B is 2.
4. The ratio of corresponding side lengths of Arrow B and Arrow A is 2 for all corresponding sides.
5. The missing side lengths in B (corresponding to 1, 2, 6 in A) are 2, 4, 12 respectively.