Quiz on dilations in geometry, including a graph and multiple-choice questions about scale factors and coordinate changes.

Here are the step-by-step solutions for the questions shown in the image.

Question 1

Problem: Choose the correct scale factor for the dilation shown in the graph.

Step-by-Step Reasoning:
1. Identify corresponding points: Look at point $P$ on the smaller triangle and point $P'$ on the larger triangle.
* Point $P$ is at coordinates $(-2, 2)$.
* Point $P'$ is at coordinates $(-6, 6)$.
2. Calculate the scale factor: The scale factor ($k$) is found by dividing a coordinate of the image (the new shape) by the corresponding coordinate of the pre-image (the original shape).
* Using the x-coordinates: $\frac{-6}{-2} = 3$
* Using the y-coordinates: $\frac{6}{2} = 3$
3. Verify with another point: Let's check point $M$ and $M'$.
* Point $M$ is at $(2, -2)$.
* Point $M'$ is at $(6, -6)$.
* Scale factor: $\frac{6}{2} = 3$ and $\frac{-6}{-2} = 3$.

The scale factor is 3. This matches option D.

Question 2

Problem: The point $A (8, 12)$ was dilated to become point $A' (2, 3)$. What was the scale factor?

Step-by-Step Reasoning:
1. Formula: Scale Factor = $\frac{\text{Image Coordinate}}{\text{Original Coordinate}}$
2. Calculate using x-coordinates:
* Original $x = 8$, Image $x' = 2$.
* Scale Factor = $\frac{2}{8} = \frac{1}{4}$
3. Check using y-coordinates:
* Original $y = 12$, Image $y' = 3$.
* Scale Factor = $\frac{3}{12} = \frac{1}{4}$

The scale factor is $\frac{1}{4}$. This matches option A.

Question 3

Problem: A dilation is which of the following?

Step-by-Step Reasoning:
1. Definition: A dilation is a transformation that changes the size of a figure but not its shape.
2. Types of Dilations:
* If the scale factor is greater than 1, it is an enlargement.
* If the scale factor is between 0 and 1, it is a reduction.
3. Conclusion: Since a dilation can be either bigger or smaller depending on the number used, it is either an enlargement or a reduction.

This matches option D.

Question 4

Problem: State the coordinate of the image of the given point $B (4,9)$ under a dilation with center at the origin with the given scale factor of 2.

Step-by-Step Reasoning:
1. Rule: To dilate a point centered at the origin $(0,0)$, you multiply both the x and y coordinates by the scale factor ($k$).
* Formula: $(x, y) \rightarrow (k \cdot x, k \cdot y)$
2. Apply the values:
* Original Point: $(4, 9)$
* Scale Factor: $2$
* New x-coordinate: $4 \times 2 = 8$
* New y-coordinate: $9 \times 2 = 18$
3. Result: The new point is $(8, 18)$.

This matches option B.

Question 5

Problem: When $A(3, 6)$ is under a dilation of 2 what are the coordinates of $A'$?

Step-by-Step Reasoning:
1. Rule: Multiply the original coordinates by the scale factor.
2. Apply the values:
* Original Point: $(3, 6)$
* Scale Factor: $2$
* New x-coordinate: $3 \times 2 = 6$
* New y-coordinate: $6 \times 2 = 12$
3. Result: The new point is $(6, 12)$.

This matches option B.

Question 6

Problem: A dilation produces a congruent figure. (True/False implied)

Step-by-Step Reasoning:
1. Definition of Congruent: Figures are congruent if they have the exact same size and shape.
2. Effect of Dilation: A dilation changes the size of the figure (unless the scale factor is exactly 1). Because the size changes, the figures are similar, but not congruent.
3. Conclusion: The statement "A dilation produces a congruent figure" is False.

*(Note: Although there are no multiple-choice options visible for #6, the standard answer to this statement is False.)*

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Final Answer:
1. D (3)
2. A (¼)
3. D (Either an enlargement or a reduction)
4. B (B' (8, 18))
5. B ((6, 12))
6. False