Exercises: Verify dilation properties - Common Core Math - HS Geometry

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

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

A.

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

B.

A transformation always preserves the distance between points.

C.

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

D.

A transformation always maps a figure to a congruent figure.

2.

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?

A.

53.5 km

B.

175 km

C.

14.3 km

D.

350 km

3.

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

A.

Rigid motions change both the size and shape of figures.

B.

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

C.

Rigid motions preserve both the size and shape of figures.

D.

Rigid motions always move figures to a different location.

B

Fluency Practice

Apply the properties of dilations directly. Show your reasoning.

1.

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 $A'B'$ ?

2.

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 $C$ ?

A.

16 units — the image is farther from the center.

B.

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

C.

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

D.

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

3.

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 $\ell'$ ?

4.

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 $m$ ?

A.

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

B.

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

C.

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

D.

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

5.

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?

C

Mixed Practice

These problems test the same skills in different formats.

D

Word Problems

Read each problem carefully and apply properties of dilations to solve.

1.

An architect draws a blueprint of a room. On the blueprint, the room measures 4 cm by 6 cm. The blueprint uses a scale factor of $k = 50$ (1 cm on the blueprint represents 50 cm in the real room).

What is the actual length of the longer side of the room in centimeters?

2.

On a coordinate grid, segment $EF$ has endpoints $E(2, 1)$ and $F(6, 1)$ . A dilation with center $O(0, 0)$ and scale factor $k = \frac{3}{2}$ maps $E \to E'$ and $F \to F'$ .

1.

What is the length of the image segment $E'F'$ ?

2.

Segment $EF$ lies on a horizontal line. After the dilation, what is true about the line containing $E'F'$ ?

A.

It is perpendicular to $EF$ because the dilation rotates lines.

B.

It is parallel to $EF$ because $EF$ does not pass through center $O$ .

C.

It is the same line as $EF$ because $EF$ is horizontal.

D.

It is the same line as $EF$ because both segments have the same slope.

E

Find the Mistake

Each problem shows a student's incorrect claim. Identify the error.

1.

Priya dilates line $\ell$ (which has slope 3 and does not pass through center $C$ ) with scale factor $k = 2$ . She concludes:

"After the dilation, the image line $\ell'$ is perpendicular to $\ell$ because dilating a line changes its direction."

What is Priya's error?

A.

Priya is correct — a dilation with $k = 2$ rotates lines by 90°.

B.

Priya confused dilations with rotations. A dilation preserves direction (slope), so $\ell'$ is parallel to $\ell$ , not perpendicular.

C.

Priya should have said the image is the same line as $\ell$ .

D.

Priya's error is that the center of dilation also moves during the transformation.

2.

Kenji performs a dilation with $k = 3$ on triangle $PQR$ and gets triangle $P'Q'R'$ . He then states:

"A dilation is just like a translation — both transformations move the figure without changing it. Since a translation is a rigid motion, so is a dilation."

What is the flaw in Kenji's reasoning?

A.

Kenji is correct — dilations and translations are both rigid motions.

B.

Kenji confuses rigid motions with similarity transformations. A translation preserves all distances; a dilation with $k \neq 1$ changes distances. The image triangle $P'Q'R'$ has sides three times as long as $PQR$ — the figures are similar, not congruent.

C.

Kenji's error is that only the vertices of the triangle were dilated, not the sides.

D.

Kenji's error is that a dilation with $k = 3$ makes the figure three times smaller.

F

Challenge Problems

These are harder problems that extend your thinking.

1.

After a dilation centered at the origin, the image of point $A$ is $A'(6, 9)$ . The scale factor is $k = 3$ . What are the coordinates of the pre-image point $A$ ?

2.