Question

warm-up

1. determine if the two regular pentagons below are proportional.
2. the distance between home and school is 4.5 miles. on a map it shows the distance to be 9 cm. what is the scale?

Explanation:

1. Determine if the two regular pentagons are proportional

Step1: Find the ratio of corresponding sides

For the side - lengths given, find the ratio of the shorter sides and the ratio of the longer sides. Let the first pentagon have side - lengths \(a_1 = 4\) in and \(b_1=2.8\) in, and the second pentagon have side - lengths \(a_2 = 14\) in and \(b_2 = 9.8\) in.
The ratio of the longer sides is \(\frac{a_2}{a_1}=\frac{14}{4}=\frac{7}{2}=3.5\).
The ratio of the shorter sides is \(\frac{b_2}{b_1}=\frac{9.8}{2.8}=\frac{98}{28}=\frac{7}{2}=3.5\).

Step2: Check for proportionality

Since the ratios of the corresponding sides are equal (\(\frac{a_2}{a_1}=\frac{b_2}{b_1} = 3.5\)), the two regular pentagons are proportional.

Step1: Convert miles to centimeters

We know that 1 mile = 160934.4 cm. So, 4.5 miles is \(4.5\times160934.4=724204.8\) cm.

Step2: Calculate the scale

The scale of a map is the ratio of the distance on the map to the actual distance. The distance on the map is \(d_{map}=9\) cm and the actual distance is \(d_{actual}=724204.8\) cm.
The scale is \(\frac{d_{map}}{d_{actual}}=\frac{9}{724204.8}=\frac{1}{80467.2}\). So the scale is \(1:80467.2\).

Answer:

Yes

2. Find the scale of the map