Back to Exercise: Verify dilation properties

Exercises: Properties of Dilations

Work through each section in order. Show your work where indicated.

Grade 9·20 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-srt-a-1

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Warm-Up: Review What You Know

These problems review skills you have already learned.

A transformation maps each point in the plane to exactly one output point. Which statement best describes this?

A transformation is a function from the plane to the plane.

A transformation always preserves the distance between points.

A transformation must be one of translation, rotation, or reflection.

A transformation always maps a figure to a congruent figure.

A map uses a scale of 1 cm = 50 km. On the map, two cities are 3.5 cm apart. What is the actual distance between the cities?

53.5 km

175 km

14.3 km

350 km

Which of the following is true about rigid motions (translations, rotations, and reflections)?

Rigid motions change both the size and shape of figures.

Rigid motions change the shape of figures but preserve their size.

Rigid motions preserve both the size and shape of figures.

Rigid motions always move figures to a different location.

Fluency Practice

Apply the properties of dilations directly. Show your reasoning.

$Coordinate plane showing triangle ABC and its dilation A'B'C' with center at the origin and scale factor 2. Dashed rays connect the center through each pair of corresponding vertices.$

Triangle $ABC$ has vertices $A(1, 1)$ , $B(3, 1)$ , and $C(1, 3)$ . A dilation with center $O(0, 0)$ and scale factor $k = 2$ maps each vertex to $A' = (2, 2)$ , $B' = (6, 2)$ , and $C' = (2, 6)$ . What is the length of side $A'B'$ ?

A dilation has center $C$ and scale factor $k = 0.5$ . Point $P$ is 8 units from $C$ . After the dilation, how far is $P'$ from $C$ ?

16 units — the image is farther from the center.

4 units — the image is half as far from the center.

8 units — the center and image stay the same distance apart.

0.5 units — the scale factor is subtracted from the distance.

$Two parallel lines with slope 2. The original line ell is closer to center C; the image line ell-prime is farther, shifted by the dilation but maintaining the same slope.$

Line $\ell$ has slope $2$ and does not pass through center $C$ . After a dilation with center $C$ and scale factor $k = 3$ , what is the slope of the image line $\ell'$ ?

Line $m$ passes through the center $C$ of a dilation with scale factor $k = 5$ . After the dilation, which best describes the image of line $m$ ?

The image is a new line parallel to $m$ .

The image is a new line perpendicular to $m$ .

The image is the same line $m$ — the line is unchanged.

The image does not exist because lines through the center are fixed points.

Segment $PQ$ has length 9 cm. It is dilated with scale factor $k = \frac{2}{3}$ . What is the length of the image segment $P'Q'$ in centimeters?

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