Math 2 Quiz: Using Similarity For Side Lengths
14 questions · exam conditions
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Using Similarity For Side LengthsQuestion 1 of 14

Triangles JKL and MNP are similar. In triangle JKL, JK = 8, KL = 12, and JL = 10. In triangle MNP, MN = 20. If vertex J corresponds to vertex M, what is the perimeter of triangle MNP?

60 units
75 units
90 units
100 units
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Math 2 Quiz

Math 2 Quiz: Using Similarity For Side Lengths

Practice Using Similarity For Side Lengths in Math 2 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 Using Similarity For Side Lengths, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

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.

All questions

Question 1

Triangles JKL and MNP are similar. In triangle JKL, JK = 8, KL = 12, and JL = 10. In triangle MNP, MN = 20. If vertex J corresponds to vertex M, what is the perimeter of triangle MNP?

  1. 60 units
  2. 75 units (correct answer)
  3. 90 units
  4. 100 units
Explanation: Since J corresponds to M, side JK corresponds to side MN. The ratio is JK:MN = 8:20 = 2:5. All corresponding sides have this ratio. So KL corresponds to NP: 12 × (5/2) = 30, and JL corresponds to MP: 10 × (5/2) = 25. Perimeter of MNP = 20 + 30 + 25 = 75. Choice A uses ratio 2:1 instead of 2:5. Choice C incorrectly triples each side. Choice D uses an incorrect scaling factor.

Question 2

Triangle ABC is similar to triangle DEF with a ratio of similarity of 3:5. If the perimeter of triangle ABC is 42 units and side DE corresponds to side AB, what is the length of side DE if AB = 15 units?

  1. 25 units (correct answer)
  2. 9 units
  3. 21 units
  4. 35 units
Explanation: Since the triangles are similar with ratio 3:5, corresponding sides are in the ratio 3:5. If AB = 15 and corresponds to DE, then AB/DE = 3/5, so 15/DE = 3/5. Cross-multiplying: 3·DE = 15·5 = 75, so DE = 25. Choice B uses the inverse ratio incorrectly (15 × 3/5 = 9). Choice C incorrectly adds the ratio difference to AB (15 + 6 = 21). Choice D incorrectly uses the perimeter relationship (42 × 5/6 ≈ 35).

Question 3

Two similar triangular gardens have perimeters of 36 feet and 48 feet respectively. If the smaller garden requires 12 bags of mulch to cover completely, how many bags of mulch will the larger garden require?

  1. 16 bags
  2. 18 bags
  3. 21.33 bags (correct answer)
  4. 24 bags
Explanation: The ratio of perimeters is 36:48 = 3:4. For similar figures, areas are in the ratio of the squares of corresponding linear measurements. So area ratio = (3:4)² = 9:16. Since mulch coverage is proportional to area, the larger garden needs 12 × (16/9) = 192/9 ≈ 21.33 bags. Choice A uses the linear ratio (12 × 4/3). Choice B incorrectly adds 6 to 12. Choice D doubles the original amount.

Question 4

A model airplane has a wingspan of 24 inches and is built to a scale of 1:48 compared to the actual airplane. If the actual airplane's fuselage is 576 inches long, what is the length of the model's fuselage?

  1. 12 inches (correct answer)
  2. 15 inches
  3. 18 inches
  4. 24 inches
Explanation: The scale 1:48 means model measurements are 1/48 of actual measurements. The model fuselage length = actual fuselage length ÷ 48 = 576 ÷ 48 = 12 inches. Choice B incorrectly uses a different ratio (576 ÷ 38.4). Choice C uses an incorrect scale calculation (576 ÷ 32). Choice D assumes the fuselage equals the wingspan without applying the scale factor.

Question 5

A scale drawing of a building has a height of 8 inches and represents an actual building height of 120 feet. On the same drawing, a window measures 0.5 inches wide. What is the actual width of the window in feet?

  1. 5.75 feet
  2. 6.25 feet
  3. 8.5 feet
  4. 7.5 feet (correct answer)
Explanation: Scale drawing problems test your ability to work with proportional relationships. When you see a question involving a scale drawing, you need to establish the scale ratio and then apply it consistently to find unknown measurements. First, establish the scale ratio using the given information. The drawing height is 8 inches and represents 120 feet of actual height. This gives us the ratio: 8 inches120 feet=1 inch15 feet\frac{8 \text{ inches}}{120 \text{ feet}} = \frac{1 \text{ inch}}{15 \text{ feet}} Now apply this same ratio to find the actual window width. If the window measures 0.5 inches on the drawing, then: 0.5 inches×15 feet per inch=7.5 feet0.5 \text{ inches} \times 15 \text{ feet per inch} = 7.5 \text{ feet} This confirms answer D is correct. Let's examine why the other answers are wrong. Answer A (5.75 feet) might result from incorrectly calculating the scale ratio or making arithmetic errors in the conversion. Answer B (6.25 feet) could come from mixing up which measurement goes in the numerator versus denominator when setting up the proportion. Answer C (8.5 feet) might occur if you accidentally used the drawing's building height (8 inches) in your calculation instead of the window width (0.5 inches). The key strategy for scale drawing problems is to always establish your scale ratio first using the complete information given, then apply that exact same ratio to find the unknown measurement. Double-check that your ratio makes sense – in this case, 1 drawing inch represents 15 actual feet, which is a reasonable scale for an architectural drawing.

Question 6

A photograph measuring 4 inches by 6 inches is enlarged so that the longer side becomes 15 inches. If the original photo and enlargement are similar rectangles, what is the area of the enlarged photograph?

  1. 90 square inches
  2. 150 square inches (correct answer)
  3. 135 square inches
  4. 225 square inches
Explanation: When you encounter problems involving similar rectangles, remember that corresponding sides are proportional and you can use scale factors to find unknown measurements. Since the rectangles are similar, the ratio between corresponding sides must be constant. The original photo's longer side is 6 inches, and it enlarges to 15 inches. This gives us a scale factor of 156=2.5\frac{15}{6} = 2.5. Apply this same scale factor to the shorter side: 4×2.5=104 \times 2.5 = 10 inches. So the enlarged photo measures 10 inches by 15 inches, giving an area of 10×15=15010 \times 15 = 150 square inches. Looking at the wrong answers: Choice A (90 square inches) would result from incorrectly keeping one dimension unchanged (6 × 15 = 90). Choice C (135 square inches) might come from averaging the original and some incorrect enlarged dimensions. Choice D (225 square inches) equals 15², suggesting the misconception that both dimensions become 15 inches, ignoring the rectangular shape. Notice that you could also solve this using the area scale factor: when linear dimensions are scaled by factor 2.5, area scales by (2.5)2=6.25(2.5)^2 = 6.25. The original area is 4×6=244 \times 6 = 24 square inches, so the enlarged area is 24×6.25=15024 \times 6.25 = 150 square inches. Study tip: For similar figures, always identify the scale factor first by comparing corresponding sides, then apply it consistently to all linear measurements. Remember that area scales by the square of the linear scale factor.

Question 7

Two similar right triangles have corresponding altitudes in the ratio 4:94:9. If the area of the smaller triangle is 32 square units, what is the area of the larger triangle?

  1. 72 square units
  2. 128 square units
  3. 162 square units (correct answer)
  4. 288 square units
Explanation: When triangles are similar, the ratio of their areas equals the square of the ratio of corresponding linear measurements. Since the altitudes are in ratio 4:9, the areas are in ratio 4²:9² = 16:81. If the smaller triangle has area 32, then the larger triangle has area 32 × (81/16) = 32 × 5.0625 = 162 square units. Choice A incorrectly uses the linear ratio instead of the squared ratio. Choice B doubles the smaller area and adds some arbitrary factor. Choice D uses the ratio 9:4 instead of 4:9.

Question 8

A rectangular garden plot is enlarged by a scale factor of 43\frac{4}{3}. If the original plot had an area of 54 square meters, what is the area of the enlarged plot?

  1. 72 square meters
  2. 81 square meters
  3. 96 square meters (correct answer)
  4. 108 square meters
Explanation: When a figure is scaled by a factor of 4/3, its area is scaled by the square of that factor: (4/3)² = 16/9. The new area is 54 × (16/9) = 54 × 16/9 = 96 square meters. Choice A incorrectly uses the linear scale factor instead of its square (54 × 4/3 = 72). Choice B uses an incorrect calculation. Choice D doubles the original area incorrectly.

Question 9

A scale model of a building is constructed where 2 centimeters represents 5 meters. If a window on the model measures 1.2 cm by 0.8 cm, what are the actual dimensions of the window in meters?

  1. 3.0 m by 2.0 m (correct answer)
  2. 2.4 m by 1.6 m
  3. 4.8 m by 3.2 m
  4. 6.0 m by 4.0 m
Explanation: Scale problems test your ability to set up and solve proportional relationships. When you see a scale model question, you need to convert between model measurements and real-world measurements using the given ratio. The scale tells you that 2 cm on the model represents 5 meters in reality. To find the actual window dimensions, you need to set up proportions for each measurement. For the length (1.2 cm): 2 cm5 m=1.2 cmx m\frac{2 \text{ cm}}{5 \text{ m}} = \frac{1.2 \text{ cm}}{x \text{ m}} Cross-multiplying: 2x=1.2×5=62x = 1.2 \times 5 = 6, so x=3.0 mx = 3.0 \text{ m} For the width (0.8 cm): 2 cm5 m=0.8 cmy m\frac{2 \text{ cm}}{5 \text{ m}} = \frac{0.8 \text{ cm}}{y \text{ m}} Cross-multiplying: 2y=0.8×5=42y = 0.8 \times 5 = 4, so y=2.0 my = 2.0 \text{ m} The actual window dimensions are 3.0 m by 2.0 m, which is answer A. Let's examine why the other answers are wrong. Answer B (2.4 m by 1.6 m) results from multiplying the model dimensions by 2 instead of using the proper scale factor of 2.5. Answer C (4.8 m by 3.2 m) comes from multiplying by 4, perhaps confusing the scale ratio. Answer D (6.0 m by 4.0 m) results from multiplying by 5, using only part of the scale relationship. For scale problems, always identify the scale factor first. Here, since 2 cm = 5 m, the scale factor is 5÷2 = 2.5. Multiply each model dimension by 2.5 to get the real dimensions quickly.

Question 10

Two similar polygons have corresponding sides in the ratio 5:75:7. If the sum of all sides (perimeter) of the smaller polygon is 45 units, and one side of the larger polygon measures 21 units, what is the length of the corresponding side in the smaller polygon?

  1. 15 units (correct answer)
  2. 12 units
  3. 18 units
  4. 25 units
Explanation: When you encounter problems involving similar polygons, remember that all corresponding measurements maintain the same ratio. This includes sides, perimeters, diagonals, and any other linear measurements. Since the polygons are similar with corresponding sides in ratio 5:75:7, their perimeters also have this same ratio. If the smaller polygon's perimeter is 45 units, you can find the larger polygon's perimeter: 45x=57\frac{45}{x} = \frac{5}{7}, so x=63x = 63 units. Now, knowing one side of the larger polygon is 21 units, you can find its corresponding side in the smaller polygon using the ratio. Since the ratio is 5:75:7 (smaller to larger), the corresponding side in the smaller polygon is: 57×21=15\frac{5}{7} \times 21 = 15 units. Looking at the wrong answers: B) 12 units results from incorrectly using 47×21\frac{4}{7} \times 21, possibly from misremembering the ratio. C) 18 units comes from using 67×21\frac{6}{7} \times 21, another ratio error. D) 25 units appears if you mistakenly multiply 21×5721 \times \frac{5}{7} but flip the fraction to 75\frac{7}{5}, giving you the side if the larger polygon corresponded to the smaller one. The correct answer is A) 15 units. Strategy tip: Always write down the ratio clearly (smaller:larger or larger:smaller) and stick to it consistently. Double-check by verifying that your answer maintains the same ratio relationship throughout the problem.

Question 11

Triangle ABCABC has sides of length 5, 12, and 13. Triangle DEFDEF is similar to triangle ABCABC with the shortest side of triangle DEFDEF measuring 15 units. What is the length of the longest side of triangle DEFDEF?

  1. 36 units
  2. 65 units
  3. 45 units
  4. 39 units (correct answer)
Explanation: When you encounter similar triangles, the key principle is that all corresponding sides are proportional by the same scale factor. This means if you know how one side changes, you can determine how all other sides change. First, identify the sides of triangle ABCABC: 5, 12, and 13 units. Notice that 52+122=25+144=169=1325^2 + 12^2 = 25 + 144 = 169 = 13^2, confirming this is a right triangle. The shortest side is 5 units and the longest is 13 units. Since triangle DEFDEF is similar to triangle ABCABC with its shortest side measuring 15 units, you can find the scale factor: 155=3\frac{15}{5} = 3. This means every side of triangle DEFDEF is exactly 3 times the corresponding side of triangle ABCABC. Therefore, the longest side of triangle DEFDEF is 13×3=3913 \times 3 = 39 units. Let's examine why the other answers are incorrect: Choice A (36 units) results from incorrectly multiplying the middle side length: 12×3=3612 \times 3 = 36. Choice B (65 units) comes from adding the scale factor instead of multiplying: 13+3=1613 + 3 = 16, though this doesn't even match 65 exactly. Choice C (45 units) appears to use an incorrect scale factor of 153=5\frac{15}{3} = 5, then multiplying 9×59 \times 5 (using wrong side lengths entirely). Strategy tip: In similar triangle problems, always establish the scale factor first by comparing corresponding sides, then apply that same factor to find any unknown side. Double-check by ensuring your scale factor works consistently across all given information.

Question 12

Two similar pentagons have areas in the ratio 4:25. If the smaller pentagon has a side length of 6 cm, what is the corresponding side length of the larger pentagon?

  1. 37.5 cm
  2. 12.5 cm
  3. 15 cm (correct answer)
  4. 18.75 cm
Explanation: When you encounter similar polygons with given area ratios, remember that areas scale with the square of corresponding linear dimensions, while side lengths scale directly with the linear scale factor. Since the pentagons are similar with areas in the ratio 4:25, you need to find the linear scale factor first. Because area scales as the square of linear dimensions, if the linear scale factor is kk, then the area ratio is k2k^2. Here, k2=254k^2 = \frac{25}{4}, so k=52=2.5k = \frac{5}{2} = 2.5. This means the larger pentagon's corresponding side length is 2.5 times the smaller pentagon's side length: 6 cm×2.5=15 cm6 \text{ cm} \times 2.5 = 15 \text{ cm}. Looking at the wrong answers: Choice A (37.5 cm) results from multiplying 6 by 6.25, which would happen if you mistakenly used the area ratio (25/4 = 6.25) directly instead of taking its square root. Choice B (12.5 cm) comes from adding the area ratio to the original side length (6 + 6.25 = 12.25, rounded), which has no geometric meaning. Choice D (18.75 cm) results from multiplying by approximately 3.125, which might come from incorrectly calculating the scale factor. The key strategy for similar figure problems: always remember that linear dimensions (like side lengths, perimeters, heights) scale by the linear scale factor kk, while areas scale by k2k^2 and volumes scale by k3k^3. When given an area ratio, take the square root to find the linear scale factor.

Question 13

Quadrilateral MNPQ is similar to quadrilateral RSTU with a ratio of 4:7. If the sum of two corresponding sides in MNPQ is 24 units, what is the sum of the corresponding sides in RSTU?

  1. 56 units
  2. 48 units
  3. 36 units
  4. 42 units (correct answer)
Explanation: When you encounter similar figures with given ratios, you're working with proportional relationships. The key insight is that if two figures are similar, ALL corresponding linear measurements (sides, perimeters, diagonals) maintain the same ratio. Given that quadrilateral MNPQ is similar to quadrilateral RSTU with a ratio of 4:7, this means every corresponding side in RSTU is 74\frac{7}{4} times the length of the corresponding side in MNPQ. Since the sum of two corresponding sides in MNPQ is 24 units, you can find the sum of the corresponding sides in RSTU by applying this same ratio: 24×74=1684=4224 \times \frac{7}{4} = \frac{168}{4} = 42 units. Let's examine why the other answers are incorrect. Choice A (56 units) results from incorrectly multiplying 24 by 73\frac{7}{3} instead of 74\frac{7}{4}. Choice B (48 units) comes from simply doubling 24, which ignores the given ratio entirely. Choice C (36 units) results from multiplying by 32\frac{3}{2}, which reverses the relationship and uses an incorrect ratio. The correct answer is D) 42 units. Study tip: When working with similar figures, remember that the ratio applies to ALL corresponding linear measurements equally. If you know the ratio and one measurement, multiply by the ratio to find the corresponding measurement in the similar figure. Always double-check that you're using the ratio in the correct direction (smaller to larger or vice versa).

Question 14

A rectangular photograph measuring 6 inches by 9 inches is enlarged so that its new width is 10 inches. If the enlarged photograph maintains the same proportions, what is the new length?

  1. 13 inches
  2. 15 inches (correct answer)
  3. 16 inches
  4. 18 inches
Explanation: The original rectangle has dimensions 6 by 9 inches. When enlarged to width 10 inches, the scale factor is 10/6 = 5/3. The new length will be 9 × (5/3) = 15 inches. Choice A incorrectly adds the difference (10-6=4) to the original length (9+4=13). Choice C uses an incorrect scale factor calculation. Choice D assumes the length doubles when width increases.