Question

1. if two circles have radii ( r_1 = 4 ) and ( r_2 = 12 ), what is the scale factor needed to map the smaller circle onto the larger circle?

options: a. 3, b. 2, c. 1.5, d. 2.5

1. what does the ratio in a scale, such as ( 1 , \text{cm} : 500 , \text{m} ), represent?

options: a. the distance between two points on a map, b. the angle measures in a scale drawing, c. the relationship between the scale drawing and the actual dimensions, d. the size of the preimage

1. if the scale factor of a dilation is ( \frac{1}{8} ), which of the following describes the result?

options: a. the image is a reduction of the preimage, b. the image is larger than the preimage, c. the preimage is a reduction of the image, d. the image is congruent to the preimage

Explanation:

Question 10

Step1: Recall scale factor formula

To map the smaller circle to the larger one, the scale factor \( k \) is the ratio of the radius of the larger circle to the smaller one. So \( k=\frac{r_2}{r_1} \).

Step2: Substitute values

Given \( r_1 = 4 \) and \( r_2=12 \), then \( k=\frac{12}{4}=3 \).

A scale ratio like \( 1\space\text{cm}:500\space\text{m} \) shows how the length on the drawing (scale drawing) relates to the actual length (actual dimensions). Option a is about map distance, not the ratio's meaning. Option b is about angles, which scales don't represent. Option d is about preimage size, not the ratio. So the ratio represents the relationship between scale drawing and actual dimensions.

A dilation scale factor of \( \frac{1}{2} \) (less than 1) means the image is smaller than the preimage, so it's a reduction of the preimage. Option b is wrong as \( \frac{1}{2}<1 \), image is smaller. Option c is wrong as preimage is larger. Option d is wrong as congruent needs scale factor 1.

Answer:

a. 3

Question 11