Question

1. a rectangle has vertices (6, 4), (2, 4), (6, -2), and (2, -2). what are the coordinates of the vertices of the image after a dilation with the origin as its center and a scale factor of 1.5?
2. what is the scale factor of the dilation? is it an enlargement or a reduction?
3. a rectangle on the coordinate plane has vertices at (0, 0), (3, 0), (3, 2), and (0, 2). a dilation of the rectangle has vertices at (0, 0), (9, 0), (9, 6), and (0, 6). find the scale factor.

Explanation:

Problem 9

Step1: Dilate each vertex by 1.5

For a point $(x,y)$ with dilation center origin and scale factor $k$, the image is $(k \cdot x, k \cdot y)$.

• $(6,4)$: $(1.5 \times 6, 1.5 \times 4) = (9, 6)$
• $(2,4)$: $(1.5 \times 2, 1.5 \times 4) = (3, 6)$
• $(6,-2)$: $(1.5 \times 6, 1.5 \times -2) = (9, -3)$
• $(2,-2)$: $(1.5 \times 2, 1.5 \times -2) = (3, -3)$

Problem 10

Step1: Identify corresponding points

Take point $M(-4, 2)$ and $M'(-2, 4)$.

Step2: Calculate scale factor

Scale factor $k = \frac{\text{image }x\text{-coordinate}}{\text{pre-image }x\text{-coordinate}} = \frac{-2}{-4} = 2$.
Since $k>1$, it is an enlargement.

Problem 11

Step1: Use corresponding vertices

Take non-origin point $(3,0)$ and its image $(9,0)$.

Step2: Compute scale factor

$k = \frac{9}{3} = 3$.

Answer:

1. The image vertices are $(9, 6)$, $(3, 6)$, $(9, -3)$, $(3, -3)$
2. Scale factor = 2; it is an enlargement
3. Scale factor = 3