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).

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.

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 $$x$$ dilated by scale factor $$r$$ produces length $$y$$, so: $$y = rx$$. This means $$r = \frac{y}{x}$$.

Now, when the same original segment undergoes dilation with scale factor $$3r$$, the new length becomes: $$x \cdot(3r) = 3rx$$. Since we know that $$rx = y$$, we can substitute: $$3rx = 3y$$.

Therefore, the resulting segment has length $$3y$$.

Looking at the wrong answers: Choice A ($$y + 3$$) incorrectly adds the scale factor instead of multiplying, which isn't how dilations work. Choice B ($$y + 3r$$) makes the same addition error while also including the variable $$r$$, but dilation requires multiplication of the original length. Choice C ($$\frac{y}{3}$$) 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 $$k$$" means "multiply all lengths by $$k$$." 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.

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.

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 $$s$$. After the first dilation by scale factor $$\frac{2}{3}$$, the new side length becomes $$s \cdot \frac{2}{3} = \frac{2s}{3}$$.

Next, this resulting square is dilated by scale factor $$m$$, giving a final side length of $$\frac{2s}{3} \cdot m = \frac{2sm}{3}$$.

We're told this final side length equals $$\frac{4}{3}$$ times the original side length, so: $$\frac{2sm}{3} = \frac{4s}{3}$$

Solving for $$m$$: multiply both sides by $$\frac{3}{2s}$$ to get $$m = \frac{4s}{3} \cdot \frac{3}{2s} = \frac{4}{2} = 2$$.

Looking at the wrong answers: Choice A ($$\frac{8}{9}$$) likely comes from incorrectly multiplying the two scale factors together. Choice B ($$\frac{4}{3}$$) represents the ratio of final to original side length, not the second scale factor. Choice C ($$\frac{3}{2}$$) 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.

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).

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).

This problem examines dilations with scale factor k = 1/2, which creates a reduction. When a figure is dilated, every segment length is multiplied by the scale factor to produce the corresponding image segment length. For quadrilateral PQRS dilated to P'Q'R'S', each image segment equals the original segment times 1/2. This means P'Q' = (1/2) × PQ, Q'R' = (1/2) × QR, and so on for all sides. Therefore, each image segment is 1/2 the length of the corresponding original segment, confirming answer A. Students might confuse this with doubling (choice B), but a scale factor less than 1 always produces a smaller image. Remember: multiply original lengths by the scale factor to find image lengths in any dilation.

This problem involves a dilation with scale factor k = 2, which creates an enlargement. In dilations, the scale factor determines how segment lengths change: each image segment equals the original segment multiplied by the scale factor. For triangle GHI dilated to G'H'I', we have G'H' = 2 × GH, H'I' = 2 × HI, and G'I' = 2 × GI. This means each image side is 2 times the length of the corresponding original side, confirming answer B. Students might mistakenly think k = 2 means adding 2 units (choice D), but dilations multiply lengths, not add to them. The key strategy is to recognize that scale factor k means "multiply all lengths by k" to find the image measurements.

This problem examines dilations with scale factor k = 4/3, which produces an enlargement. When a rectangle is dilated, every side length is multiplied by the scale factor to create the corresponding image side. For rectangle LMNO dilated to L'M'N'O', each image side equals the original side times 4/3. This means L'M' = (4/3) × LM, M'N' = (4/3) × MN, and so on, confirming that each image side is 4/3 times the length of the corresponding original side (answer A). Students might confuse this with the reciprocal 3/4 (choice B), but k = 4/3 means multiply by 4/3, not 3/4. The misconception in choice D treats dilation as addition rather than multiplication. To solve dilation problems correctly, always multiply lengths by the given scale factor.