Math 1 Quiz: Dilation And Similarity
20 questions · exam conditions
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Dilation And SimilarityQuestion 1 of 20

Rectangle DEFGDEFG is dilated with center at the origin and scale factor 33 to produce rectangle D′E′F′G′D'E'F'G'. If the diagonal of rectangle DEFGDEFG has length 1010 units, what is the ratio of the area of rectangle D′E′F′G′D'E'F'G' to the area of rectangle DEFGDEFG?

3:13:1
6:16:1
9:19:1
30:130:1
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Math 1 Quiz

Math 1 Quiz: Dilation And Similarity

Practice Dilation And Similarity in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Dilation And Similarity, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

Rectangle DEFGDEFG is dilated with center at the origin and scale factor 33 to produce rectangle D′E′F′G′D'E'F'G'. If the diagonal of rectangle DEFGDEFG has length 1010 units, what is the ratio of the area of rectangle D′E′F′G′D'E'F'G' to the area of rectangle DEFGDEFG?

  1. 3:13:1
  2. 6:16:1
  3. 9:19:1 (correct answer)
  4. 30:130:1
Explanation: When a figure is dilated by scale factor k, its area is multiplied by k². Since the scale factor is 3, the area ratio is 3² = 9:1. Choice A gives the linear scale factor ratio. Choice B incorrectly doubles the scale factor. Choice D incorrectly multiplies the scale factor by the original diagonal length.

Question 2

In the coordinate plane, point M(4,−6)M(4, -6) is dilated with center at the origin and scale factor 34\frac{3}{4} to produce point M′M'. Point N(a,b)N(a, b) is then dilated with the same center and scale factor to produce point N′(6,−9)N'(6, -9). What are the coordinates of point NN?

  1. (4.5,−6.75)(4.5, -6.75)
  2. (8,−12)(8, -12) (correct answer)
  3. (2.25,−3.375)(2.25, -3.375)
  4. (4,−6)(4, -6)
Explanation: To find N, we reverse the dilation of N' by dividing by the scale factor: N = N' ÷ (3/4) = N' × (4/3). So N = (6 × 4/3, -9 × 4/3) = (8, -12). Choice A incorrectly applies the scale factor 3/4 to N'. Choice C incorrectly uses (3/4)² as the factor. Choice D incorrectly assumes N = M.

Question 3

Triangle JKLJKL undergoes two successive dilations: first with scale factor 12\frac{1}{2} centered at point AA, then with scale factor 44 centered at point BB. If the original triangle has a side length of 1212 units, what is the corresponding side length in the final image?

  1. 66 units
  2. 2424 units (correct answer)
  3. 4848 units
  4. 9696 units
Explanation: Successive dilations combine by multiplying their scale factors: (1/2) × 4 = 2. The final side length is 12 × 2 = 24 units. Choice A uses only the first dilation. Choice C incorrectly adds the scale factors (1/2 + 4 = 4.5, then 12 × 4 = 48). Choice D uses 12 × (1/2) × 4 × 4 incorrectly.

Question 4

Parallelogram RSTURSTU undergoes a dilation with scale factor k>1k > 1 to produce parallelogram R′S′T′U′R'S'T'U'. If the ratio of the area of R′S′T′U′R'S'T'U' to the area of RSTURSTU is 4916\frac{49}{16}, and angle R=110°R = 110°, what is the measure of angle R′R' and the value of kk?

  1. Angle R′=110°R' = 110°, k=74k = \frac{7}{4} (correct answer)
  2. Angle R′=4916×110°R' = \frac{49}{16} \times 110°, k=74k = \frac{7}{4}
  3. Angle R′=110°R' = 110°, k=4916k = \frac{49}{16}
  4. Angle R′=74×110°R' = \frac{7}{4} \times 110°, k=4916k = \frac{49}{16}
Explanation: Dilation preserves angles, so angle R' = 110°. Since area scales by k², we have k² = 49/16, so k = 7/4 (taking the positive value since k > 1). Choice B incorrectly scales the angle by the area ratio. Choice C uses the area ratio as the linear scale factor. Choice D incorrectly scales the angle by k and uses the area ratio as k.

Question 5

Triangle MNOMNO has vertices M(0,4)M(0, 4), N(6,0)N(6, 0), and O(−3,−2)O(-3, -2). After a dilation with center at the origin, the image triangle M′N′O′M'N'O' has vertex M′(0,10)M'(0, 10). What are the coordinates of vertex N′N'?

  1. (15,0)(15, 0) (correct answer)
  2. (6,10)(6, 10)
  3. (10,4)(10, 4)
  4. (12,6)(12, 6)
Explanation: The scale factor is found using M and M': since M(0,4) maps to M'(0,10), the scale factor is 10/4 = 5/2. Applying this to N(6,0): N' = (6 × 5/2, 0 × 5/2) = (15, 0). Choice B incorrectly keeps the x-coordinate unchanged. Choice C swaps and incorrectly scales coordinates. Choice D uses scale factor 2 instead of 5/2.

Question 6

Quadrilateral PQRSPQRS is similar to quadrilateral WXYZWXYZ with a scale factor of 2:52:5. If quadrilateral PQRSPQRS is dilated to become congruent to quadrilateral WXYZWXYZ, and the area of the dilated quadrilateral is 100100 square units, what was the area of the original quadrilateral PQRSPQRS?

  1. 1616 square units (correct answer)
  2. 2525 square units
  3. 4040 square units
  4. 6464 square units
Explanation: The scale factor from PQRS to the dilated quadrilateral is 5/2. Since area scales by the square of the linear scale factor, the area ratio is (5/2)² = 25/4. If the dilated area is 100, then the original area is 100 ÷ (25/4) = 100 × (4/25) = 16 square units. Choice B uses the linear scale factor incorrectly. Choice C uses scale factor 5/2 instead of (5/2)². Choice D uses (2/5)² incorrectly applied.

Question 7

Circle AA has radius 66 and circle BB has radius 1515. If circle AA is dilated with center at point CC to produce a circle similar to circle BB, and the distance from CC to the center of circle AA is 88, what is the distance from CC to the center of the dilated circle?

  1. 1212 units
  2. 2020 units (correct answer)
  3. 2323 units
  4. 3232 units
Explanation: The scale factor needed is 15/6 = 5/2. Under dilation, distances from the center of dilation are multiplied by the scale factor. So the distance becomes 8 × (5/2) = 20 units. Choice A incorrectly uses scale factor 6/15. Choice C incorrectly adds the original distance to the radius of circle B. Choice D incorrectly multiplies by 4 instead of 5/2.

Question 8

Regular hexagon ABCDEFABCDEF with side length ss is dilated with center at its centroid and scale factor 23\frac{2}{3} to produce hexagon A′B′C′D′E′F′A'B'C'D'E'F'. If the distance from the centroid to vertex AA in the original hexagon is dd, what is the distance from vertex A′A' to the corresponding vertex in the original hexagon?

  1. d3\frac{d}{3} (correct answer)
  2. 2d3\frac{2d}{3}
  3. s3\frac{s}{3}
  4. d−2d3d - \frac{2d}{3}
Explanation: Under dilation with center at the centroid and scale factor 2/3, vertex A moves to A' such that the distance from centroid to A' is (2/3)d. Since A, centroid, and A' are collinear with A' between the centroid and A, the distance AA' = d - (2/3)d = d/3. Choice B gives the distance from centroid to A'. Choice C incorrectly uses the side length. Choice D gives the calculation but not the simplified result.

Question 9

Pentagon ABCDEABCDE is similar to pentagon FGHIJFGHIJ. If pentagon ABCDEABCDE can be transformed into pentagon FGHIJFGHIJ by a dilation with scale factor kk followed by a rotation, and the perimeter of ABCDEABCDE is 3030 while the perimeter of FGHIJFGHIJ is 4545, what is the value of kk?

  1. 23\frac{2}{3}
  2. 49\frac{4}{9}
  3. 32\frac{3}{2} (correct answer)
  4. 94\frac{9}{4}
Explanation: When you encounter problems involving similar figures and transformations, remember that dilations change the size of figures while preserving their shape. The key insight is understanding how scale factors relate to linear measurements like perimeter. In a dilation with scale factor kk, every linear dimension of the original figure is multiplied by kk. Since perimeter is a linear measurement (sum of side lengths), the perimeter of the dilated figure equals the original perimeter times kk. Here, pentagon ABCDEABCDE is transformed into pentagon FGHIJFGHIJ through dilation followed by rotation. Since rotation doesn't change size, only the dilation affects the perimeter. If the scale factor is kk, then: Perimeter of FGHIJ=k×Perimeter of ABCDE\text{Perimeter of } FGHIJ = k \times \text{Perimeter of } ABCDE Substituting the given values: 45=k×3045 = k \times 30 k=4530=32k = \frac{45}{30} = \frac{3}{2} So the correct answer is C. Looking at the wrong answers: A) 23\frac{2}{3} is the reciprocal of the correct answer—this would happen if you incorrectly calculated 3045\frac{30}{45} instead of 4530\frac{45}{30}. B) 49\frac{4}{9} is (23)2\left(\frac{2}{3}\right)^2, which might result from confusing linear scale factors with area scale factors. D) 94\frac{9}{4} is (32)2\left(\frac{3}{2}\right)^2, another area-related confusion. Remember: for similar figures, the ratio of corresponding linear measurements (like perimeters) equals the scale factor, while the ratio of areas equals the scale factor squared. Don't mix these up!

Question 10

Circle PP with center (3,−1)(3, -1) and radius 44 is dilated with center (7,2)(7, 2) and scale factor 32\frac{3}{2} to produce circle P′P'. What are the center and radius of circle P′P'?

  1. Center (−1,−5.5)(-1, -5.5), radius 66
  2. Center (10.5,3)(10.5, 3), radius 66
  3. Center (4.5,−1.5)(4.5, -1.5), radius 66
  4. Center (1,−2.5)(1, -2.5), radius 66 (correct answer)
Explanation: When you encounter a dilation problem, you need to understand how transformations affect both the position and size of geometric figures. A dilation moves every point along a ray from the center of dilation and scales distances by the given factor. To find the new center of circle P′P', you apply the dilation formula. The center of dilation is (7,2)(7, 2), the original center is (3,−1)(3, -1), and the scale factor is 32\frac{3}{2}. First, find the vector from the dilation center to the original center: (3−7,−1−2)=(−4,−3)(3-7, -1-2) = (-4, -3). Then multiply this vector by the scale factor: 32⋅(−4,−3)=(−6,−4.5)\frac{3}{2} \cdot (-4, -3) = (-6, -4.5). Finally, add this result to the dilation center: (7,2)+(−6,−4.5)=(1,−2.5)(7, 2) + (-6, -4.5) = (1, -2.5). The radius simply gets multiplied by the scale factor: 4⋅32=64 \cdot \frac{3}{2} = 6. Choice A gives (−1,−5.5)(-1, -5.5) for the center, which suggests subtracting the scaled vector instead of adding it to the dilation center. Choice B gives (10.5,3)(10.5, 3), which appears to add the original center coordinates to the dilation center incorrectly. Choice C gives (4.5,−1.5)(4.5, -1.5), which looks like it used the wrong calculation for the vector transformation. The correct answer is D: center (1,−2.5)(1, -2.5), radius 66. Remember that dilations always preserve the center-to-center direction from the dilation point, but scale the distance. Always work systematically: find the vector, scale it, then translate from the dilation center.

Question 11

Triangle ABCABC undergoes a dilation with center PP and scale factor k=23k = \frac{2}{3} to produce triangle A′B′C′A'B'C'. If the perimeter of triangle ABCABC is 2424 units and angle B=65°B = 65°, what is the measure of angle B′B' in triangle A′B′C′A'B'C'?

  1. 65°65° (correct answer)
  2. 23⋅65°=43.33°\frac{2}{3} \cdot 65° = 43.33°
  3. 32⋅65°=97.5°\frac{3}{2} \cdot 65° = 97.5°
  4. 90°−65°=25°90° - 65° = 25°
Explanation: Under dilation, angles are preserved (remain unchanged) while lengths are scaled by the scale factor. Since angle B = 65° in the original triangle, angle B' = 65° in the dilated triangle. Choice B incorrectly applies the scale factor to the angle measure. Choice C incorrectly applies the reciprocal of the scale factor. Choice D incorrectly assumes some complementary angle relationship.

Question 12

Triangle ABC is dilated by a scale factor of 34\frac{3}{4} about point P to create triangle A'B'C'. If the perimeter of triangle ABC is 48 units and angle B measures 65°, what is the measure of angle B' in triangle A'B'C' and what is the perimeter of triangle A'B'C'?

  1. Angle B' = 65°, perimeter = 36 units (correct answer)
  2. Angle B' = 48.75°, perimeter = 36 units
  3. Angle B' = 65°, perimeter = 64 units
  4. Angle B' = 48.75°, perimeter = 64 units
Explanation: Under dilation, angles are preserved (remain unchanged) while lengths are multiplied by the scale factor. Therefore, angle B' = 65° (same as angle B). The perimeter is multiplied by the scale factor: 48 × (3/4) = 36 units. Choice B incorrectly applies the scale factor to the angle (65 × 3/4 = 48.75°). Choice C incorrectly adds instead of multiplying for perimeter (48 + 16 = 64). Choice D makes both errors.

Question 13

Rectangle DEFG has vertices at D(2, 4), E(8, 4), F(8, 10), and G(2, 10). If this rectangle is dilated by a scale factor of 2.5 about the origin, which statement about the resulting rectangle D'E'F'G' is correct?

  1. The area of D'E'F'G' is 2.5 times the area of DEFG, and the angles remain 90°
  2. The area of D'E'F'G' is 6.25 times the area of DEFG, and the angles remain 90° (correct answer)
  3. The area of D'E'F'G' is 2.5 times the area of DEFG, and the angles become 225°
  4. The area of D'E'F'G' is 6.25 times the area of DEFG, and the angles become 225°
Explanation: Under dilation, angles are preserved, so all angles remain 90°. Linear dimensions are multiplied by the scale factor (2.5), but area is multiplied by the square of the scale factor: (2.5)² = 6.25. Choice A incorrectly uses the linear scale factor for area. Choices C and D incorrectly calculate angle measures as 90° × 2.5 = 225°, but angles are preserved under dilation.

Question 14

Circle O has radius 12 units and is dilated by scale factor kk to produce circle O'. If the ratio of the circumference of circle O' to the circumference of circle O is 56\frac{5}{6}, and both circles undergo the same dilation again, what will be the radius of the final image of circle O?

  1. 253\frac{25}{3} units (correct answer)
  2. 10 units
  3. 256\frac{25}{6} units
  4. 506\frac{50}{6} units
Explanation: Since circumference = 2πr, the ratio of circumferences equals the ratio of radii, which equals the scale factor k. So k = 5/6. After the first dilation, circle O' has radius 12 × (5/6) = 10 units. After the second dilation with the same scale factor, the final radius is 10 × (5/6) = 50/6 = 25/3 units. Choice B gives the radius after one dilation only (10 units). Choice C incorrectly calculates 12 × (5/6) ÷ 2. Choice D gives 50/6 without simplifying to 25/3.

Question 15

Triangle ABC has been dilated to form triangle DEF. The ratio of the perimeter of triangle ABC to the perimeter of triangle DEF is 3:7. If the altitude from vertex A to side BC has length 15 units, what is the length of the corresponding altitude from vertex D to side EF?

  1. 457\frac{45}{7} units
  2. 21 units
  3. 35 units (correct answer)
  4. 1053\frac{105}{3} units
Explanation: When figures are similar (which dilated figures always are), the ratio of all corresponding linear measurements equals the ratio of perimeters. Since the perimeter ratio is 3:7, the altitude ratio is also 3:7. If the altitude in ABC is 15 units, then the corresponding altitude in DEF is 15 × (7/3) = 35 units. Choice A incorrectly calculates 15 × (3/7) = 45/7. Choice B uses an additive relationship (15 + 6 = 21). Choice D calculates 105/3 = 35 but expresses it as an improper fraction, which is unnecessarily complex.

Question 16

Two similar polygons have corresponding sides in the ratio 4:9. If a diagonal of the smaller polygon measures 20 units, and the difference between the areas of the two polygons is 260 square units, what is the length of the corresponding diagonal in the larger polygon?

  1. 35 units
  2. 65 units
  3. 55 units
  4. 45 units (correct answer)
Explanation: When you encounter similar polygons, remember that all corresponding linear measurements (sides, diagonals, heights) are in the same ratio, while areas are in the ratio of the square of that linear ratio. Given that corresponding sides are in the ratio 4:9, all linear measurements follow this same ratio. If the smaller polygon's diagonal is 20 units, then the larger polygon's diagonal is 94×20=45\frac{9}{4} \times 20 = 45 units. Let's verify this using the area information. Since areas of similar figures are in the ratio of the square of their linear ratio, the area ratio is 42:92=16:814^2:9^2 = 16:81. If the smaller polygon has area AA, the larger has area 81A16\frac{81A}{16}. The difference is 81A16−A=65A16=260\frac{81A}{16} - A = \frac{65A}{16} = 260, giving us A=64A = 64 square units. This confirms our polygons exist and our ratio is correct. Looking at the wrong answers: A) 35 units incorrectly adds the ratio difference (5) to 30, showing confusion about proportional relationships. B) 65 units might come from misusing the area difference of 260 in the calculation. C) 55 units could result from incorrectly applying the ratio as 114\frac{11}{4} instead of 94\frac{9}{4}. Study tip: In similar polygon problems, always apply the given ratio directly to find corresponding linear measurements first. Use area relationships as a check, remembering that area ratios are the square of linear ratios. Don't let complex area calculations distract you from the simpler proportional reasoning.

Question 17

A regular hexagon undergoes a dilation with scale factor 23\frac{2}{3}. If the original hexagon has a side length of 12 units, which of the following statements about the relationship between the original hexagon and its image is true?

  1. The hexagons are congruent because dilation preserves both size and shape
  2. The hexagons are congruent because the scale factor creates an equivalent transformation
  3. The hexagons are neither congruent nor similar because the scale factor is not equal to 1
  4. The hexagons are similar because dilation preserves shape but changes size proportionally (correct answer)
Explanation: When you encounter questions about dilations, focus on how transformations affect the geometric relationships between figures. Dilations are transformations that resize figures by a scale factor while keeping the same center point. In this problem, the regular hexagon undergoes a dilation with scale factor 23\frac{2}{3}, which means every dimension of the original figure is multiplied by 23\frac{2}{3}. The new side length becomes 12×23=812 \times \frac{2}{3} = 8 units. Since the scale factor is not equal to 1, the image is smaller than the original, but all angles remain unchanged and all sides are proportionally reduced. This creates similar figures. Similarity means the shapes have identical angles and proportional corresponding sides. Dilations always preserve shape while changing size proportionally, making the original and image similar regardless of the scale factor (unless the scale factor is 1, which would also make them congruent). Option A is wrong because dilation changes size—it only preserves shape, not both size and shape. Option B incorrectly suggests the transformation creates congruent figures; while dilations are indeed valid transformations, a scale factor of 23\frac{2}{3} changes the size. Option C contains a critical misconception—figures can absolutely be similar when the scale factor isn't 1. In fact, that's exactly when you get similarity without congruence. Remember this key distinction: congruent figures are identical in size and shape, while similar figures have identical shape but proportional sizes. Any dilation with scale factor ≠ 1 creates similar (but not congruent) figures.

Question 18

Two similar rectangles have areas in the ratio 16:2516:25. If the smaller rectangle is dilated to have the same area as the larger rectangle, what is the scale factor of this dilation?

  1. 45\frac{4}{5}
  2. 2516\frac{25}{16}
  3. 1625\frac{16}{25}
  4. 54\frac{5}{4} (correct answer)
Explanation: When you encounter problems involving similar figures and area ratios, remember that area scales as the square of the linear scale factor. If two similar rectangles have a linear scale factor of kk, their areas will have a ratio of k2k^2. Since the rectangles have areas in the ratio 16:2516:25, you need to find what linear scale factor would transform the smaller rectangle to match the larger one's area. First, find the current linear scale factor between the rectangles. Since area ratio equals (linear scale factor)2(\text{linear scale factor})^2, you have k2=1625k^2 = \frac{16}{25}, so k=45k = \frac{4}{5}. This means the smaller rectangle's dimensions are 45\frac{4}{5} times those of the larger rectangle. To dilate the smaller rectangle so its area matches the larger rectangle's area, you need to find what factor will make their areas equal. If you multiply the smaller rectangle's area by some factor f2f^2, you want: 16⋅f2=2516 \cdot f^2 = 25, so f2=2516f^2 = \frac{25}{16} and f=54f = \frac{5}{4}. Choice A (45\frac{4}{5}) represents the current linear scale factor between the rectangles, not the dilation needed. Choice B (2516\frac{25}{16}) gives you f2f^2, the area scale factor, rather than the linear scale factor ff. Choice C (1625\frac{16}{25}) is the reciprocal of the area scale factor, pointing in the wrong direction. Study tip: Always distinguish between linear scale factors and area scale factors. When dilating to achieve equal areas, work with f2f^2 first, then take the square root to find the linear dilation factor.

Question 19

Two similar triangles have corresponding sides in the ratio 4:74:7. If the smaller triangle undergoes a dilation to become congruent to the larger triangle, what is the scale factor of this dilation?

  1. 47\frac{4}{7}
  2. 74\frac{7}{4} (correct answer)
  3. 37\frac{3}{7}
  4. 73\frac{7}{3}
Explanation: Since the triangles have corresponding sides in ratio 4:7, to make the smaller triangle congruent to the larger one, each side of the smaller triangle must be multiplied by 7/4. This is the scale factor of the required dilation. Choice A would shrink the smaller triangle further. Choice C represents the difference in ratios incorrectly calculated. Choice D uses the difference 7-4=3 incorrectly.

Question 20

Point Q(6,8)Q(6, 8) is dilated with center (2,3)(2, 3) and scale factor 12\frac{1}{2} to produce point Q′Q'. What are the coordinates of Q′Q'?

  1. (3,4)(3, 4)
  2. (1,1.5)(1, 1.5)
  3. (5,7)(5, 7)
  4. (4,5.5)(4, 5.5) (correct answer)
Explanation: When you encounter a dilation problem, you're working with a transformation that changes the size of a figure while maintaining its shape. The key is understanding that dilation moves each point along the line connecting it to the center of dilation, scaling the distance by the given factor. To find Q′Q', you need to apply the dilation formula. First, find the vector from the center (2,3)(2, 3) to point Q(6,8)Q(6, 8): (6−2,8−3)=(4,5)(6-2, 8-3) = (4, 5). This represents the displacement from center to original point. Next, multiply this displacement vector by the scale factor 12\frac{1}{2}: 12(4,5)=(2,2.5)\frac{1}{2}(4, 5) = (2, 2.5). This gives you the new displacement from center to the dilated point. Finally, add this scaled displacement to the center coordinates: (2,3)+(2,2.5)=(4,5.5)(2, 3) + (2, 2.5) = (4, 5.5). Therefore, Q′Q' is at (4,5.5)(4, 5.5). Choice A (3,4)(3, 4) appears to come from incorrectly applying the scale factor directly to the original coordinates. Choice B (1,1.5)(1, 1.5) results from subtracting the scaled displacement instead of adding it. Choice C (5,7)(5, 7) likely comes from moving one unit toward the center in each direction, ignoring the proper scale factor calculation. Remember the dilation formula: new point = center + scale factor × (original point - center). Always work with the displacement from the center, not the original coordinates directly.