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Geometry

Geometry Help: Dilations Change Length By Scale Factor

Review real example questions for Dilations Change Length By Scale Factor in Geometry.

Question 1

In the coordinate plane, segment PQ has endpoints P(2, 4) and Q(8, 12). After a dilation centered at the origin, the image segment P'Q' has length 15 units. What was the scale factor of the dilation?

  1. 12\frac{1}{2}21​
  2. 32\frac{3}{2}23​
  3. 2
  4. 3
Explanation: First find the length of PQ: PQ = √[(8-2)² + (12-4)²] = √[36 + 64] = √100 = 10 units. Since P'Q' has length 15 units, the scale factor is 15/10 = 3/2. Choice A gives a length of 5 units. Choice C gives a length of 20 units. Choice D gives a length of 30 units.

Question 2

A regular hexagon has side length 12 cm. It undergoes a dilation with scale factor k, and the resulting hexagon has a perimeter that is 54 cm less than the original perimeter. What is the value of k?

  1. 14\frac{1}{4}41​
  2. 12\frac{1}{2}21​
  3. 34\frac{3}{4}43​
  4. 23\frac{2}{3}32​
Explanation: The original perimeter is 6 × 12 = 72 cm. The new perimeter is 72 - 54 = 18 cm. Since dilation multiplies all lengths by k, we have 72k = 18, so k = 18/72 = 1/4. Choice B gives a perimeter of 36 cm (difference of 36 cm). Choice C gives a perimeter of 54 cm (difference of 18 cm). Choice D gives a perimeter of 48 cm (difference of 24 cm).

Question 3

In the coordinate plane, triangle ABCABCABC is dilated about center OOO with scale factor k=32k=\tfrac{3}{2}k=23​ to form triangle A′B′C′A'B'C'A′B′C′. How does the length of segment AB‾\overline{AB}AB compare to the length of its image A′B′‾\overline{A'B'}A′B′?

  1. A′B′‾\overline{A'B'}A′B′ is 23\tfrac{2}{3}32​ as long as AB‾\overline{AB}AB.
  2. A′B′‾\overline{A'B'}A′B′ is 32\tfrac{3}{2}23​ as long as AB‾\overline{AB}AB.
  3. A′B′‾\overline{A'B'}A′B′ is the same length as AB‾\overline{AB}AB.
  4. A′B′‾\overline{A'B'}A′B′ is longer than AB‾\overline{AB}AB by 12\tfrac{1}{2}21​ unit.
Explanation: This question tests understanding of how dilations with scale factor k = 3/2 affect segment lengths. In a dilation, the scale factor multiplies all distances from the center, which means each segment length is multiplied by the scale factor. Since triangle ABC is dilated to form triangle A'B'C', the corresponding segments are AB and A'B'. Applying the scale factor k = 3/2, we get A'B' = (3/2) × AB, which means A'B' is 3/2 as long as AB. This matches answer choice B, confirming that the image segment is 1.5 times the original length. A common misconception (choice A) is to think the reciprocal 2/3 applies, but dilations multiply lengths by the scale factor, not its reciprocal. To solve dilation problems, always multiply the original length by the scale factor to find the image length.

Question 4

A line segment of length x undergoes a dilation with scale factor r to produce a segment of length y. If the same original segment undergoes a dilation with scale factor 3r, what will be the length of the resulting segment in terms of y?

  1. y+3y + 3y+3
  2. y+3ry + 3ry+3r
  3. y3\frac{y}{3}3y​
  4. 3y3y3y
Explanation: When you encounter dilation problems, remember that dilation creates similar figures by multiplying all corresponding lengths by the same scale factor. The key insight is understanding how scale factors relate to each other. Let's establish the relationship from the given information. The original segment of length xxx dilated by scale factor rrr produces length yyy, so: y=rxy = rxy=rx. This means r=yxr = \frac{y}{x}r=xy​. Now, when the same original segment undergoes dilation with scale factor 3r3r3r, the new length becomes: x⋅(3r)=3rxx \cdot (3r) = 3rxx⋅(3r)=3rx. Since we know that rx=yrx = yrx=y, we can substitute: 3rx=3y3rx = 3y3rx=3y. Therefore, the resulting segment has length 3y3y3y. Looking at the wrong answers: Choice A (y+3y + 3y+3) incorrectly adds the scale factor instead of multiplying, which isn't how dilations work. Choice B (y+3ry + 3ry+3r) makes the same addition error while also including the variable rrr, but dilation requires multiplication of the original length. Choice C (y3\frac{y}{3}3y​) represents the opposite relationship—this would be the result if you divided by 3 rather than multiplied, suggesting a misunderstanding of how scale factors greater than 1 affect size. Study tip: In dilation problems, always remember that "scale factor kkk" means "multiply all lengths by kkk." When you see relationships between different scale factors applied to the same original figure, look for proportional reasoning opportunities—if one scale factor produces a certain result, a scale factor three times larger will produce a result three times larger.

Question 5

Segment JK‾\overline{JK}JK is dilated about center OOO to form J′K′‾\overline{J'K'}J′K′. The diagram shows OJ=4OJ=4OJ=4 units and OJ′=10OJ'=10OJ′=10 units. Which claim about length is supported by the diagram?

  1. J′K′J'K'J′K′ is 25\tfrac{2}{5}52​ as long as JKJKJK.
  2. J′K′J'K'J′K′ is 52\tfrac{5}{2}25​ as long as JKJKJK.
  3. J′K′J'K'J′K′ is the same length as JKJKJK.
  4. J′K′J'K'J′K′ is longer than JKJKJK by 666 units.
Explanation: This question tests finding the scale factor from given distances and applying it to segment lengths. The diagram shows OJ = 4 units and OJ' = 10 units, so the scale factor k = OJ'/OJ = 10/4 = 5/2. In a dilation, this scale factor applies to all segments, not just distances from the center. Therefore, J'K' = (5/2) × JK, which means J'K' is 5/2 as long as JK, confirming answer B. This represents an enlargement where the image is 2.5 times the original length. A common error (choice A) is to use the reciprocal 2/5, but the scale factor is always the ratio of image distance to original distance from the center. Remember: once you find the scale factor from any corresponding distances, it applies to all segment lengths in the figure.

Question 6

A square with side length s is dilated by scale factor 23\frac{2}{3}32​. The resulting square is then dilated by scale factor m. If the final square has side length equal to 43\frac{4}{3}34​ times the original side length s, what is the value of m?

  1. 89\frac{8}{9}98​
  2. 43\frac{4}{3}34​
  3. 32\frac{3}{2}23​
  4. 2
Explanation: When you encounter problems involving multiple transformations, the key is to track how each transformation affects the dimensions step by step, then work backward from the final result. Let's trace what happens to the square's side length through both dilations. The original square has side length sss. After the first dilation by scale factor 23\frac{2}{3}32​, the new side length becomes s⋅23=2s3s \cdot \frac{2}{3} = \frac{2s}{3}s⋅32​=32s​. Next, this resulting square is dilated by scale factor mmm, giving a final side length of 2s3⋅m=2sm3\frac{2s}{3} \cdot m = \frac{2sm}{3}32s​⋅m=32sm​. We're told this final side length equals 43\frac{4}{3}34​ times the original side length, so: 2sm3=4s3\frac{2sm}{3} = \frac{4s}{3}32sm​=34s​ Solving for mmm: multiply both sides by 32s\frac{3}{2s}2s3​ to get m=4s3⋅32s=42=2m = \frac{4s}{3} \cdot \frac{3}{2s} = \frac{4}{2} = 2m=34s​⋅2s3​=24​=2. Looking at the wrong answers: Choice A (89\frac{8}{9}98​) likely comes from incorrectly multiplying the two scale factors together. Choice B (43\frac{4}{3}34​) represents the ratio of final to original side length, not the second scale factor. Choice C (32\frac{3}{2}23​) is the reciprocal of the first scale factor, suggesting confusion about which transformation to undo. Remember: when multiple transformations are applied sequentially, set up equations that track the cumulative effect, then solve for the unknown parameter. Don't assume you need to multiply or divide the given scale factors directly.

Question 7

Parallelogram JKLM undergoes a dilation with center at vertex J and scale factor 35\frac{3}{5}53​. In the original parallelogram, JK = 25 units and JM = 15 units. What is the perimeter of the image parallelogram J'K'L'M'?

  1. 72 units
  2. 56 units
  3. 48 units
  4. 80 units
Explanation: When you encounter a dilation problem, remember that dilation scales all lengths by the same factor while preserving shape. The key insight is that if each side length changes by the scale factor, the perimeter changes by that same factor. In the original parallelogram JKLM, you're given JK = 25 units and JM = 15 units. Since opposite sides of a parallelogram are equal, the four sides are: JK = 25, KL = 15, LM = 25, and MJ = 15. The original perimeter is 25+15+25+15=8025 + 15 + 25 + 15 = 8025+15+25+15=80 units. Under a dilation with scale factor 35\frac{3}{5}53​, each side length gets multiplied by 35\frac{3}{5}53​. The new side lengths become: 25×35=1525 \times \frac{3}{5} = 1525×53​=15 and 15×35=915 \times \frac{3}{5} = 915×53​=9. So the image parallelogram has sides of 15, 9, 15, and 9 units, giving a perimeter of 15+9+15+9=4815 + 9 + 15 + 9 = 4815+9+15+9=48 units. Choice A (72 units) represents the error of multiplying the original perimeter by 910\frac{9}{10}109​ instead of 35\frac{3}{5}53​. Choice B (56 units) might result from incorrectly calculating the new side lengths or adding them wrong. Choice D (80 units) is the original perimeter—this would be your answer if you forgot that dilation changes all measurements. Study tip: For any dilation, the perimeter of the image equals the original perimeter times the scale factor. This saves time: 80×35=4880 \times \frac{3}{5} = 4880×53​=48 units directly.

Question 8

A rectangle undergoes two successive dilations. First, it is dilated by a scale factor of 2, then the resulting rectangle is dilated by a scale factor of 13\frac{1}{3}31​. If the original rectangle had a diagonal of length 5 units, what is the length of the diagonal after both transformations?

  1. 56\frac{5}{6}65​ units
  2. 103\frac{10}{3}310​ units
  3. 53\frac{5}{3}35​ units
  4. 152\frac{15}{2}215​ units
Explanation: Successive dilations multiply their scale factors: 2 × (1/3) = 2/3. The diagonal length becomes 5 × (2/3) = 10/3 units. Choice A incorrectly subtracts scale factors (2 - 1/3 = 5/3, then 5 × 1/3 = 5/3, then divides by 3). Choice C uses only the second scale factor (5 × 1/3). Choice D incorrectly adds scale factors (2 + 1/3 = 7/3, then 5 × 7/3 ÷ 2).

Question 9

Triangle ABC is dilated by a scale factor of 34\frac{3}{4}43​ to create triangle A'B'C'. If the perimeter of triangle ABC is 24 units and side AB has length 8 units, what is the length of side A'B' in triangle A'B'C'?

  1. 6 units
  2. 8 units
  3. 18 units
  4. 32 units
Explanation: In a dilation, all lengths are multiplied by the scale factor. Side AB has length 8 units, so A'B' = 8 × (3/4) = 6 units. Choice B incorrectly assumes the length stays the same. Choice C incorrectly applies the scale factor to the perimeter instead (24 × 3/4 = 18). Choice D incorrectly uses the reciprocal scale factor (8 × 4 = 32).

Question 10

Circle O has radius 8 inches. After a dilation, the image circle O' has a circumference that is 34\frac{3}{4}43​ the circumference of the original circle. What is the radius of circle O'?

  1. 102310\frac{2}{3}1032​ inches
  2. 323\frac{32}{3}332​ inches
  3. 6 inches
  4. 6136\frac{1}{3}631​ inches
Explanation: When you encounter dilation problems involving circles, remember that dilation affects all linear measurements by the same scale factor, including radius and circumference. Let's work through this step-by-step. The original circle O has radius 8 inches, so its circumference is C=2πr=2π(8)=16πC = 2\pi r = 2\pi(8) = 16\piC=2πr=2π(8)=16π inches. After dilation, circle O' has a circumference that is 34\frac{3}{4}43​ of the original, which means C′=34×16π=12πC' = \frac{3}{4} \times 16\pi = 12\piC′=43​×16π=12π inches. Since C′=2πr′C' = 2\pi r'C′=2πr′, we can solve for the new radius: 12π=2πr′12\pi = 2\pi r'12π=2πr′, so r′=6r' = 6r′=6 inches. This confirms answer C is correct. Now let's examine why the other options are wrong. Choice A (102310\frac{2}{3}1032​ inches) and choice B (323\frac{32}{3}332​ inches, which equals 102310\frac{2}{3}1032​) both give a radius larger than the original 8 inches. This is impossible since the circumference decreased, meaning the radius must also decrease. These likely come from incorrectly taking the reciprocal 43\frac{4}{3}34​ instead of 34\frac{3}{4}43​. Choice D (6136\frac{1}{3}631​ inches) is close to the correct answer but represents a computational error, possibly from incorrectly calculating 34×8\frac{3}{4} \times 843​×8 instead of properly using the circumference relationship. Remember: in dilation problems, always check whether your answer makes logical sense. If the image is smaller than the original, all corresponding measurements should be smaller too.