Dilations and Similarity Transformations: Geometry Math Worksheet – PDF Download

Problem 1: Scale factor from distances

A dilation with center O maps point P to point \(P^{\prime}\). In the figure, \(OP=6\) and \(OP^{\prime}=15\). Find the scale factor and tell whether the dilation is an enlargement or a reduction.

Problem 2: Find the image length

Segment AB is 12 inches long. After a dilation with scale factor \(\frac{3}{4}\), how long is \(A^{\prime}B^{\prime}\)?

Problem 3: Find the original length

After a dilation with scale factor 2.5, a segment measures 20 centimeters. How long was the segment before the dilation?

Problem 4: Enlargement or reduction

Classify each scale factor as an enlargement, a reduction, or no change in size.

1. \(\frac{7}{4}\)
2. \(0.6\)
3. \(1\)
4. \(1.2\)
5. \(\frac{5}{9}\)

Problem 5: Dilate points about the origin

Each point is dilated about the origin with the given scale factor. Find the image.

1. \(P(3,-2),\ k=3\)
2. \(Q(-4,6),\ k=\frac{1}{2}\)
3. \(R(5,0),\ k=4\)
4. \(S(0,-7),\ k=\frac{3}{7}\)

Problem 6: Scale factor from coordinates

A dilation with center at the origin maps each point to its image. Find the scale factor in each case.

1. \(A(2,4)\to A^{\prime}(5,10)\)
2. \(B(6,-9)\to B^{\prime}(2,-3)\)

Problem 7: Dilate a triangle

Triangle PQR is shown. Dilate it about the origin with scale factor 2. List the coordinates of \(P^{\prime},\ Q^{\prime}\) and \(R^{\prime}\), then graph the image.

Problem 8: Shrink a rectangle

Rectangle ABCD is dilated about the origin with scale factor \(\frac{1}{2}\). Find the coordinates of the image and the dimensions of the new rectangle.

Problem 9: Center not at the origin

A dilation has center \(C(1,1)\) and scale factor 3. Find the images \(P^{\prime}\) of \(P(3,2)\) and \(Q^{\prime}\) of \(Q(2,3)\), then compare the lengths of \(PQ\) and \(P^{\prime}Q^{\prime}\).

Problem 10: Dilate a line

The line \(y=3x-2\) is dilated about the origin with scale factor 2. Write an equation of the image line.

Problem 11: Which lines stay put

A dilation about the origin has scale factor 3. For each line, decide whether the image is the same line or a different line, and write an equation of the image.

1. \(y=-3x\)
2. \(y=-3x+6\)
3. \(y=\frac{1}{2}x\)
4. \(y=\frac{1}{2}x-1\)

Problem 12: Perimeter after a dilation

A triangle has side lengths 7 in., 9 in. and 12 in. It is dilated with scale factor 2.5. Find the side lengths and the perimeter of the image, and compare the perimeters.

Problem 13: Angles after a dilation

In triangle ABC, \(m\angle A=48^{\circ}\) and \(m\angle B=67^{\circ}\). The triangle is dilated with scale factor 3 to get \(A^{\prime}B^{\prime}C^{\prime}\).

1. Find \(m\angle C\).
2. Find the three angle measures of \(A^{\prime}B^{\prime}C^{\prime}\).

Problem 14: Dilation then translation

Triangle ABC has vertices \(A(1,1),\ B(3,1),\ C(1,2)\). It is dilated about the origin with scale factor 2, and the result is translated 3 units right and 1 unit down. Find the final vertices \(A^{\prime}^{\prime},\ B^{\prime}^{\prime},\ C^{\prime}^{\prime}\).

Problem 15: Two dilations in a row

Point \(P(4,6)\) is dilated about the origin with scale factor 2. The image is then dilated about the origin with scale factor \(\frac{3}{4}\). Find the final image and the single scale factor that does the same job.

Problem 16: Third angles and similarity

In the figure, triangle PQR has angles \(52^{\circ}\) and \(71^{\circ}\), and triangle XYZ has angles \(71^{\circ}\) and \(57^{\circ}\). Find the unmarked angles and decide whether the triangles are similar. If they are, write a similarity statement.

Problem 17: Similar or not

For each pair of triangles, decide whether they must be similar. Explain using angles.

1. Triangle 1 has angles \(35^{\circ}\) and \(80^{\circ}\). Triangle 2 has angles \(80^{\circ}\) and \(65^{\circ}\).
2. Triangle 3 has angles \(30^{\circ}\) and \(60^{\circ}\). Triangle 4 has angles \(45^{\circ}\) and \(45^{\circ}\).
3. Triangle 5 is isosceles with a \(40^{\circ}\) vertex angle. Triangle 6 is isosceles with a \(70^{\circ}\) base angle.

Problem 18: Slope and length after dilation

Segment PQ has endpoints \(P(2,1)\) and \(Q(5,3)\). It is dilated about the origin with scale factor 2. Find the endpoints of \(P^{\prime}Q^{\prime}\), then compare slopes and lengths of the two segments.

Problem 19: Photo enlargement

Maya enlarges a 4 inch by 6 inch photo with a dilation of scale factor 2.5 for the school lobby. Find the dimensions of the enlargement and compare the perimeters of the two photos.

Problem 20: Projector screen

A projector throws a slide that is 8 inches wide and 6 inches tall onto a wall. The image on the wall is 6 feet wide. Find the scale factor and the height of the projected image in feet.

Problem 21: Copier reduction

Jamal reduces an 11 inch by 8.5 inch flyer on a copier set to 60%. Find the dimensions of the copy. Then find the percent setting that would return the copy to full size.

Problem 22: Dilation about a vertex

Triangle ABC has \(AB=8,\ AC=6\) and \(BC=7\). It is dilated about vertex A with scale factor 1.5 to get \(AB^{\prime}C^{\prime}\), as shown. Find \(AB^{\prime},\ AC^{\prime},\ B^{\prime}C^{\prime}\) and the perimeter of each triangle.

Problem 23: Find the center

A dilation with scale factor 2 maps \(P(3,2)\) to \(P^{\prime}(5,4)\). Find the center of the dilation, then find the image of \(Q(4,5)\).

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