Math / Geometry

QuestionIn the graph below, circle $T^{\prime}$ is the image of circle $T$ after a dilation. Radii $\overline{T U}$ and $\overline{T^{\prime} U^{\prime}}$ are also shown.
What are the scale factor and center of the dilation? Simplify your answers and write them as fractions or whole numbers. scale factor: $\square$ center of the dilation: ( $\square$ $\square$ Submit Work it out Desk 1

Studdy Solution

STEP 1

1. Circle $T$ and circle $T'$ are concentric, meaning they share the same center.
2. The center of the dilation is the origin, $(0, 0)$ .
3. The dilation involves a scale factor that relates the radii of the two circles.

STEP 2

1. Determine the original and dilated radii.
2. Calculate the scale factor.
3. Identify the center of the dilation.

STEP 3

Determine the original and dilated radii:
- The radius of circle $T$ is the distance from the origin to point $U$ , which is $(8, 0)$ . - The radius of circle $T'$ is the distance from the origin to point $U'$ , which is $(4, 6)$ .
Calculate the lengths of these radii using the distance formula:
- Radius of $T$ : $\sqrt{(8 - 0)^2 + (0 - 0)^2} = 8$ - Radius of $T'$ : $\sqrt{(4 - 0)^2 + (6 - 0)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}$

STEP 4

Calculate the scale factor:
The scale factor $k$ is the ratio of the radius of $T'$ to the radius of $T$ :
$k = \frac{\text{Radius of } T'}{\text{Radius of } T} = \frac{2\sqrt{13}}{8} = \frac{\sqrt{13}}{4}$

STEP 5

Identify the center of the dilation:
Since the circles are concentric and share the same center, the center of the dilation is the origin, $(0, 0)$ .
The scale factor is:
$\boxed{\frac{\sqrt{13}}{4}}$
The center of the dilation is:
$\boxed{(0, 0)}$

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