Dilations On The Coordinate Plane

1) A triangle has sides 3, 4, and 5. After dilation, the new triangle has sides 9, 12, and 15. What is the scale factor?

3

4

5

2

New sides 9, 12, 15 are each 3× the originals 3, 4, 5. Scale factor k = 3.

Explanation

New sides 9, 12, 15 are each 3× the originals 3, 4, 5. Scale factor k = 3.

2) The vertices of a trapezoid are (1, 2), (3, 2), (2, 4), and (4, 4). If the trapezoid is dilated by a factor of 2 about the origin, one new vertex is:

(2, 4)

(6, 8)

(4, 2)

(8, 4)

Dilate about origin by k = 2: (1,2)→(2,4), (3,2)→(6,4), (2,4)→(4,8), (4,4)→(8,8).

Explanation

Dilate about origin by k = 2: (1,2)→(2,4), (3,2)→(6,4), (2,4)→(4,8), (4,4)→(8,8).

3) Which transformation could change a triangle into a larger similar triangle?

Translation

Reflection

Dilation

Rotation

Only a dilation changes size while keeping similarity.

Explanation

Only a dilation changes size while keeping similarity.

4) A dilation centered at the origin maps (–2, 5) to (–6, 15). What is the scale factor?

2

3

–3

1/3

(–2,5)→(–6,15) multiplies both coordinates by 3.

Explanation

(–2,5)→(–6,15) multiplies both coordinates by 3.

5) A rectangle 5 by 7 is dilated to become 20 by 28. The scale factor is:

4

3

2

5

Dimensions 5→20 and 7→28 both multiply by 4.

Explanation

Dimensions 5→20 and 7→28 both multiply by 4.

6) The vertices of a triangle are (0, 0), (2, 4), and (4, 2). A dilation with scale factor 0.5 produces:

(0, 0), (1, 2), (2, 1)

(0, 0), (1, 4), (2, 2)

(0, 0), (2, 8), (4, 4)

(0, 0), (1, 1), (2, 2)

Multiply each coordinate by 0.5.

Explanation

Multiply each coordinate by 0.5.

7) Which of the following is not preserved in a dilation?

Angle measure

Proportionality of sides

Orientation of figure

Actual side length

Dilations preserve angles, proportionality, and (for k>0) orientation; actual lengths change by k.

Explanation

Dilations preserve angles, proportionality, and (for k>0) orientation; actual lengths change by k.

8) A line segment from (1, 2) to (3, 4) is dilated by scale factor 2 about the origin. The image endpoints are:

(2, 4) and (6, 8)

(0.5, 1) and (1.5, 2)

(3, 6) and (5, 8)

(2, 3) and (4, 5)

Scale endpoints by k = 2: (1,2)→(2,4), (3,4)→(6,8).

Explanation

Scale endpoints by k = 2: (1,2)→(2,4), (3,4)→(6,8).

9) A triangle has sides 5, 7, and 9. After a dilation, the new triangle has sides 2.5, 3.5, and 4.5. Which scale factor was used?

1.5

0.5

2

3

Sides are half as long: 5→2.5, 7→3.5, 9→4.5.

Explanation

Sides are half as long: 5→2.5, 7→3.5, 9→4.5.

10) A hexagon with perimeter 30 is dilated with scale factor 5. What is the new perimeter?

35

60

150

25

Perimeter scales by k: 30×5 = 150.

Explanation

Perimeter scales by k: 30×5 = 150.

11) A dilation is performed on a triangle with base 10 and height 6. The new base is 15. What is the new height?

7

8

9

12

Base 10→15 gives k = 1.5, so height = 6×1.5 = 9.

Explanation

Base 10→15 gives k = 1.5, so height = 6×1.5 = 9.

12) A rectangle with sides 6 and 10 is dilated to form a new rectangle with sides 9 and 15. The ratio 9 ÷ 6 = 15 ÷ 10 represents the:

Scale factor

Side multiplier

Angle measure

Length constant

9÷6 = 15÷10 = 1.5, which is the dilation scale factor.

Explanation

9÷6 = 15÷10 = 1.5, which is the dilation scale factor.

13) Which center of dilation leaves all points unchanged?

The centroid

The origin

The point of dilation itself

The circumcenter

Under any dilation, the center stays fixed; all other points move unless k = 1.

Explanation

Under any dilation, the center stays fixed; all other points move unless k = 1.

14) A figure has a side length of 12. After dilation, the side length is 18. The scale factor is:

2/3

1/5

3/2

4/3

k = 18÷12 = 3/2 = 1.5.

Explanation

k = 18÷12 = 3/2 = 1.5.

15) A point at (–2, 5) is dilated about the origin with a scale factor of 4. What are the new coordinates?

(–8, 20)

(–0.5, 1.25)

(2, –5)

(–6, 15)

Multiply by k = 4: (–2×4, 5×4) = (–8, 20).

Explanation

Multiply by k = 4: (–2×4, 5×4) = (–8, 20).

16) Which transformation preserves shape and size without turning or flipping (i.e., a slide)?

Translation

Dilation

Reflection

Rotation

A translation is a slide that preserves shape and size without turning or flipping.

Explanation

A translation is a slide that preserves shape and size without turning or flipping.

17) A dilation centered at the origin maps (3, –2) to (1.5, –1). The scale factor is:

0.25

0.5

2

3

(3, –2)→(1.5, –1) halves both coordinates.

Explanation

(3, –2)→(1.5, –1) halves both coordinates.

18) Which of the following is always true of dilations?

They produce congruent figures

They preserve angle measures

They preserve side lengths

They change orientation

Dilations keep angles; side lengths scale unless k = 1.

Explanation

Dilations keep angles; side lengths scale unless k = 1.

19) A triangle with perimeter 12 is dilated by a scale factor of 2. What is the new perimeter?

14

24

6

36

Perimeter scales by k: 12×2 = 24.

Explanation

Perimeter scales by k: 12×2 = 24.

20) A triangle has an area of 25. After dilation with scale factor 4, the new area is:

50

100

200

400

Area scales by k²: 25×4² = 25×16 = 400.

Explanation

Area scales by k²: 25×4² = 25×16 = 400.

Dilations on the Coordinate Plane

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