Question

1. which statement best explains why all circles are similar?

a. only circles with the same radius are similar.
b. only circles that share the same center are similar.
c. all circles can be mapped onto one another using dilations and translations.
d. all circles have equal circumference.

1. two circles have radii ( r_1 = 7 ) and ( r_2 = 21 ). what is the scale factor (( k )) for the dilation?

a. 4.25
b. 3
c. 2.5
d. 2

1. a pentagon with one vertex at ( (5, 3) ) is dilated by a scale factor of ( \frac{3}{2} ) with the center at ( (0, 0) ). what are the coordinates of the dilated vertex?

a. ( (0, 3) )
b. ( (7.5, 3) )
c. ( (8, 4) )
d. ( (7, 4.5) )

1. if a figure is dilated by a scale factor of 0.25, what happens to the distances between points in the figure?

Explanation:

Question 1

• Option a: Incorrect. All circles are similar regardless of radius (similarity doesn't require same radius, just proportionality).
• Option b: Incorrect. Center position (concentricity) isn't required for similarity; similarity is about shape, not position.
• Option c: Correct. Dilations can scale a circle to match another's size, and translations can move it, showing all circles are similar via these transformations.
• Option d: Incorrect. Circles with different radii have different circumferences ($C = 2\pi r$), so circumferences aren't equal.

Step1: Recall dilation scale factor formula

The scale factor \( k \) for dilating a circle with radius \( r_1 \) to \( r_2 \) is \( k=\frac{r_2}{r_1} \) (if dilating from \( r_1 \) to \( r_2 \)) or \( \frac{r_1}{r_2} \), but here we assume dilating the first circle to the second.

Step2: Substitute values

Given \( r_1 = 7 \), \( r_2 = 21 \), so \( k=\frac{21}{7}=3 \).

Step1: Recall dilation rule for center at (0,0)

When a point \( (x,y) \) is dilated with center \( (0,0) \) and scale factor \( k \), the new coordinates are \( (kx, ky) \).

Step2: Apply the scale factor

The original point is \( (5, 3) \) (assuming the "2" was a typo, likely 3? Wait, no, the problem says (5,2)? Wait, the scale factor is \( \frac{3}{2} \)? Wait, the user's problem: "scale factor of \( \frac{3}{2} \)"? Wait, the options: let's check. Wait, original point (5,2), scale factor \( \frac{3}{2} \)? Wait, no, maybe a typo, but let's proceed. Wait, the options: a. (0,3), b. (7.5, 3), c. (8,4), d. (7,4.5). Wait, if original is (5,2), scale factor \( \frac{3}{2} \): \( 5\times\frac{3}{2}=7.5 \), \( 2\times\frac{3}{2}=3 \). So (7.5, 3), which is option b. Wait, maybe the original point was (5,2) (maybe the "2" was correct). So:

Step1: Dilation formula for (0,0)

For a point \( (x,y) \), after dilation with scale factor \( k \) and center \( (0,0) \), new point is \( (kx, ky) \).

Step2: Substitute \( x = 5 \), \( y = 2 \), \( k=\frac{3}{2} \)

\( x' = 5\times\frac{3}{2}=7.5 \), \( y' = 2\times\frac{3}{2}=3 \). So the new point is (7.5, 3).

Answer:

c. All circles can be mapped onto one another using dilations and translations.

Question 2