Proving Similarity With Transformations

1) Which sequence proves two right triangles are similar?

Translation + Reflection

Dilation + Translation

Rotation + Reflection

Shear

Dilation changes size while keeping shape, translation slides. Together they prove similarity.

Explanation

Dilation changes size while keeping shape, translation slides. Together they prove similarity.

2) A triangle with vertices (1,1), (2,3), (3,1) is translated 4 units right. One new vertex is:

(5,1)

(1,5)

(2,7)

(7,3)

Translate 4 units right: add 4 to x. (1,1) → (5,1).

Explanation

Translate 4 units right: add 4 to x. (1,1) → (5,1).

3) Point P is located at (–3, 4). A 90° clockwise rotation about the origin sends P to:

(–4, –3)

(4, 3)

(3, –4)

(–3, –4)

90° clockwise about origin: (x, y) → (y, –x). (–3,4) → (4,3).

Explanation

90° clockwise about origin: (x, y) → (y, –x). (–3,4) → (4,3).

4) A dilation with center (0,0) and scale factor ½ maps (6, –8) to:

(3, –4)

(–3, 4)

(12, –16)

(–12, 16)

Scale by 1/2 about origin: (6, –8) → (3, –4).

Explanation

Scale by 1/2 about origin: (6, –8) → (3, –4).

5) The preimage has side lengths 5, 6, and 7. The image has 10, 12, and 14. The transformation is:

Reflection

Translation

Dilation with scale factor 2

Rotation

All sides doubled: 5→10, 6→12, 7→14.

Explanation

All sides doubled: 5→10, 6→12, 7→14.

6) Triangle A is located at (0,0), (1,0), (0,1). Triangle B is located at (0,0), (2,0), (0,2). Which transformation maps A onto B?

Dilation, scale factor 2

Translation

Reflection across y-axis

Rotation

Coordinates double: (0,0),(1,0),(0,1) → (0,0),(2,0),(0,2).

Explanation

Coordinates double: (0,0),(1,0),(0,1) → (0,0),(2,0),(0,2).

7) Which similarity transformation could map a rectangle to another rectangle with doubled side lengths?

Translation

Reflection

Dilation

Rotation

Doubling side lengths is a scale change.

Explanation

Doubling side lengths is a scale change.

8) A rotation of 270° counterclockwise is equivalent to:

90° clockwise

180° clockwise

270° clockwise

90° counterclockwise

270° counterclockwise = 90° clockwise.

Explanation

270° counterclockwise = 90° clockwise.

9) A reflection across the x-axis preserves:

Orientation

Angle measures

Location

Order of vertices

Reflections preserve lengths and angles but reverse orientation.

Explanation

Reflections preserve lengths and angles but reverse orientation.

10) A sequence of a reflection and a translation will always result in figures that are:

Congruent

Enlarged

Distorted

Similar only

Reflection and translation are isometries, so distance and angles are preserved.

Explanation

Reflection and translation are isometries, so distance and angles are preserved.

11) A pentagon is dilated by a scale factor of 4. Which statement is correct about the image?

The figure becomes a different shape

Side lengths become one-fourth as long

Perimeter does not change

Angle measures remain the same

Dilations keep angles; lengths and perimeter scale.

Explanation

Dilations keep angles; lengths and perimeter scale.

12) Which is an example of two figures that are similar but not congruent?

A rectangle 4 × 6 and a rectangle 8 × 12

Two equilateral triangles with sides 5 and 5

Two circles with radius 10

A square reflected across the y-axis

Sides are proportional (scale factor 2), so similar but not congruent.

Explanation

Sides are proportional (scale factor 2), so similar but not congruent.

13) A triangle is rotated 90° counterclockwise and then translated 3 units up. The resulting triangle is:

Not similar

Smaller than the original

Larger than the original

Congruent to the original

Rotation + translation are isometries; size and shape unchanged.

Explanation

Rotation + translation are isometries; size and shape unchanged.

14) Which is true of similarity transformations?

They always change angle measure

They preserve proportionality of sides

They change orientation only

They change length only

Similarity transformations keep angles and side ratios.

Explanation

Similarity transformations keep angles and side ratios.

15) A trapezoid has bases 6 and 10. A second trapezoid has bases 9 and 15, and their corresponding angles match. What does this prove about the trapezoids?

They are congruent

They are similar

They have the same area

They must overlap on the plane

Bases are in the same ratio 6:10 = 9:15 and angles correspond.

Explanation

Bases are in the same ratio 6:10 = 9:15 and angles correspond.

16) Which transformation does NOT change orientation?

Translation

Reflection

Rotation

Dilation

Translations do not change orientation. (Reflections reverse it; rotations and positive dilations preserve it.)

Explanation

Translations do not change orientation. (Reflections reverse it; rotations and positive dilations preserve it.)

17) Which sequence maps a figure onto a congruent figure?

Dilation + Translation

Reflection + Rotation

Dilation with factor 2

Enlargement

Both are rigid motions, so the image is congruent to the original.

Explanation

Both are rigid motions, so the image is congruent to the original.

18) Point A is located at (2, –1). After reflection across the y-axis, A is mapped to:

(–2, –1)

(2, 1)

(–2, 1)

(1, –2)

Reflect across y-axis: (x, y) → (–x, y).

Explanation

Reflect across y-axis: (x, y) → (–x, y).

19) Which transformation always preserves distance?

Enlargement

Dilation

Translation

Scaling

Translations preserve all distances and angles.

Explanation

Translations preserve all distances and angles.

20) Which set of transformations could map a triangle onto a similar triangle?

Dilation then translation

Rotation then reflection

Translation only

Reflection only

Dilation creates a similar figure; translation repositions it.

Explanation

Dilation creates a similar figure; translation repositions it.

Proving Similarity with Transformations

1) Which sequence proves two right triangles are similar?

2)

What first name or nickname would you like us to use?

2) A triangle with vertices (1,1), (2,3), (3,1) is translated 4 units right. One new vertex is:

3) Point P is located at (–3, 4). A 90° clockwise rotation about the origin sends P to:

4) A dilation with center (0,0) and scale factor ½ maps (6, –8) to:

5) The preimage has side lengths 5, 6, and 7. The image has 10, 12, and 14. The transformation is:

6) Triangle A is located at (0,0), (1,0), (0,1). Triangle B is located at (0,0), (2,0), (0,2). Which transformation maps A onto B?

7) Which similarity transformation could map a rectangle to another rectangle with doubled side lengths?

8) A rotation of 270° counterclockwise is equivalent to:

9) A reflection across the x-axis preserves:

10) A sequence of a reflection and a translation will always result in figures that are:

11) A pentagon is dilated by a scale factor of 4. Which statement is correct about the image?

12) Which is an example of two figures that are similar but not congruent?

13) A triangle is rotated 90° counterclockwise and then translated 3 units up. The resulting triangle is:

14) Which is true of similarity transformations?

15) A trapezoid has bases 6 and 10. A second trapezoid has bases 9 and 15, and their corresponding angles match. What does this prove about the trapezoids?

16) Which transformation does NOT change orientation?

17) Which sequence maps a figure onto a congruent figure?

18) Point A is located at (2, –1). After reflection across the y-axis, A is mapped to:

19) Which transformation always preserves distance?

20) Which set of transformations could map a triangle onto a similar triangle?