Dilation & Similarity - Math 2

In the coordinate plane, triangle PQR with vertices P(2, 4), Q(6, 8), and R(8, 2) is dilated about the origin. The image triangle P'Q'R' has vertex P' at (3, 6). Under this same dilation, what is the area of triangle P'Q'R' if the area of triangle PQR is 20 square units?
Answer: 45 square units The scale factor is found by comparing corresponding vertices: P(2,4) maps to P'(3,6), so k = 3/2 = 1.5. Under dilation, areas are multiplied by k² = (1.5)² = 2.25. Therefore, the area of P'Q'R' = 20 × 2.25 = 45 square units. Choice A uses the linear scale factor. Choice B uses k² = 1.75 incorrectly. Choice C uses k = 2.
A regular hexagon is dilated by scale factor k to create a similar hexagon. If the ratio of the area of the original hexagon to the area of the dilated hexagon is 16:25, and the side length of the dilated hexagon is 15 units, what is the side length of the original hexagon?
Answer: 12 units When you encounter dilation problems involving area ratios, remember that areas scale by the square of the linear scale factor. If a figure is dilated by scale factor $k$ , then the area changes by a factor of $k^2$ . Given that the ratio of original area to dilated area is 16:25, we can write: $\frac{\text{original area}}{\text{dilated area}} = \frac{16}{25}$ Since area scales by $k^2$ , this ratio equals $\frac{1}{k^2}$ . Therefore: $\frac{1}{k^2} = \frac{16}{25}$ Solving for $k$ : $k^2 = \frac{25}{16}$ , so $k = \frac{5}{4} = 1.25$ This means the dilated hexagon is 1.25 times larger than the original. Since the dilated hexagon has side length 15 units, the original side length is: $\frac{15}{1.25} = \frac{15}{\frac{5}{4}} = 15 \times \frac{4}{5} = 12$ units. Choice A (12 units) is correct. Choice B (9 units) results from incorrectly using $k = \frac{4}{5}$ instead of $\frac{5}{4}$ —confusing which hexagon is larger. Choice C (18 units) comes from incorrectly assuming the linear ratio equals the area ratio and calculating $15 \times \frac{4}{5} \times 1.5$ . Choice D (20 units) results from using $k = \frac{4}{3}$ by taking the square root of $\frac{16}{9}$ instead of $\frac{16}{25}$ . Remember: area ratios equal the square of linear scale factors. Always identify which figure is larger before setting up your proportion.
Triangle $PQR$ has vertices $P(0,0)$ , $Q(6,0)$ , and $R(3,9)$ . The triangle is dilated about point $P$ with scale factor $\frac{1}{3}$ to produce triangle $P'Q'R'$ . What are the coordinates of the centroid of triangle $P'Q'R'$ ?
Answer: $(1, 1)$ When you encounter dilation problems, remember that two key transformations occur: the coordinates change according to the scale factor, and geometric properties like centroids transform predictably. First, let's find the vertices of the dilated triangle. Since the dilation is about point $P(0,0)$ with scale factor $\frac{1}{3}$ , multiply each coordinate by $\frac{1}{3}$ : - $P'(0,0)$ (the center of dilation stays fixed) - $Q'(6 \cdot \frac{1}{3}, 0 \cdot \frac{1}{3}) = Q'(2,0)$ - $R'(3 \cdot \frac{1}{3}, 9 \cdot \frac{1}{3}) = R'(1,3)$ The centroid is the average of the three vertices' coordinates: $\text{Centroid} = \left(\frac{0+2+1}{3}, \frac{0+0+3}{3}\right) = (1,1)$ This confirms answer choice C is correct. Looking at the wrong answers: Choice A $(\frac{1}{3}, \frac{1}{3})$ results from incorrectly applying the scale factor to the original centroid $(3,3)$ after finding it, rather than dilating the triangle first. Choice B $(3,3)$ is the centroid of the original triangle—you forgot to account for the dilation entirely. Choice D $(1, \frac{1}{3})$ appears to mix correct and incorrect coordinate calculations. Strategy tip: For dilation problems, always transform the figure first, then find the requested property. Don't try to directly transform properties like centroids—work with the actual coordinates to avoid calculation errors.
In the coordinate plane, point $A$ is at $(4, -6)$ and point $B$ is at $(-2, 9)$ . Both points are dilated about the origin with scale factor $\frac{1}{2}$ to produce points $A'$ and $B'$ . What is the ratio of the length $A'B'$ to the length $AB$ ?
Answer: $\frac{1}{2}$ This question tests your understanding of how dilations affect distances between points. When you see a dilation problem, remember that the scale factor determines how all lengths in the figure change proportionally. To solve this, you need to understand that under a dilation with scale factor $k$ , every distance gets multiplied by $|k|$ . Since both points $A$ and $B$ are dilated by the same scale factor $\frac{1}{2}$ , the distance between them also gets multiplied by $\frac{1}{2}$ . Here's why: Point $A(4, -6)$ becomes $A'(2, -3)$ and point $B(-2, 9)$ becomes $B'(-1, 4.5)$ after dilation. While you could calculate both distances using the distance formula, there's a more elegant approach. Since dilation preserves ratios and angles, the triangle formed by the origin, $A$ , and $B$ is similar to the triangle formed by the origin, $A'$ , and $B'$ with a ratio of $\frac{1}{2}$ . Therefore, $\frac{A'B'}{AB} = \frac{1}{2}$ . Looking at the wrong answers: (A) gives 2, which would be the ratio if you flipped the fraction. (B) gives $\frac{1}{4}$ , which might result from incorrectly squaring the scale factor. (D) gives $\frac{\sqrt{2}}{2}$ , which has no connection to the given scale factor. Study tip: For dilation problems, remember that distances scale by the absolute value of the scale factor. You don't need to calculate actual coordinates or distances—just apply the scale factor directly to length ratios.
Triangle $JKL$ is similar to triangle $MNO$ with a ratio of similarity of $3:5$ . If the altitude from $J$ to side $KL$ in triangle $JKL$ has length 9 units, and the area of triangle $MNO$ is 200 square units, what is the length of side $KL$ ?
Answer: $16$ units The ratio of areas is $(\frac{3}{5})^2 = \frac{9}{25}$ . So the area of triangle $JKL$ is $200 \cdot \frac{9}{25} = 72$ square units. Using the area formula: $\frac{1}{2} \cdot KL \cdot 9 = 72$ , so $KL = 16$ units. Choice A comes from incorrect area calculation. Choice B uses the wrong altitude ratio. Choice C results from using linear scale factor instead of area scale factor.
Two circles are similar (one is a dilation of the other) with radii in the ratio $2:7$ . If the smaller circle has an area of $16\pi$ square units, what is the circumference of the larger circle?
Answer: $28\pi$ units The smaller circle has area $16\pi = \pi r^2$ , so $r = 4$ . Since the radii are in ratio $2:7$ , the larger circle has radius $4 \cdot \frac{7}{2} = 14$ . The circumference is $2\pi(14) = 28\pi$ . Choice A gives the radius times $\pi$ . Choice C squares the radius incorrectly. Choice D doubles the area instead of finding circumference.
Triangle $DEF$ undergoes a dilation with center $P$ and scale factor $k > 1$ to produce triangle $D'E'F'$ . Point $M$ is the midpoint of side $DE$ , and point $M'$ is the corresponding point on side $D'E'F'$ . If $PM = 6$ units and $PM' = 15$ units, what is the value of $k$ ?
Answer: $\frac{5}{2}$ In a dilation with center $P$ and scale factor $k$ , every point moves to a position where its distance from $P$ is multiplied by $k$ . Since $M$ maps to $M'$ , we have $PM' = k \cdot PM$ . Therefore $15 = k \cdot 6$ , so $k = \frac{15}{6} = \frac{5}{2}$ . Choice B incorrectly uses $k = \frac{9}{6} = \frac{3}{2}$ . Choice C represents a common arithmetic error. Choice D gives the ratio without simplifying.
A square with side length 12 units is dilated by a scale factor of $\frac{2}{3}$ about its center. A second dilation with scale factor $k$ about the same center returns the square to its original size. What is the value of $k$ ?
Answer: $\frac{3}{2}$ After the first dilation, the square has side length $12 \cdot \frac{2}{3} = 8$ . To return to the original size of 12, we need $8k = 12$ , so $k = \frac{12}{8} = \frac{3}{2}$ . Alternatively, since dilations compose by multiplying scale factors, we need $\frac{2}{3} \cdot k = 1$ , giving $k = \frac{3}{2}$ . Choice B is $(\frac{2}{3})^2$ . Choice C is $(\frac{3}{2})^2$ . Choice D assumes the scale factor should be the reciprocal of the original side length ratio.
A regular hexagon undergoes a dilation with scale factor $\frac{3}{4}$ . If the original hexagon has a side length of 8 units, what is the ratio of the area of the original hexagon to the area of the dilated hexagon?
Answer: $\frac{16}{9}$ Under dilation with scale factor $\frac{3}{4}$ , areas scale by the square of the scale factor: $(\frac{3}{4})^2 = \frac{9}{16}$ . This means the dilated hexagon has area $\frac{9}{16}$ times the original area. Therefore, the ratio of original to dilated area is $\frac{16}{9}$ . Choice A gives the linear scale factor. Choice B gives the reciprocal of the linear scale factor. Choice C gives the area scale factor (dilated to original) instead of original to dilated.
Two similar triangles have corresponding altitudes of 9 inches and 15 inches. If the triangles are related by a dilation and the sum of their perimeters is 96 inches, what is the perimeter of the larger triangle?
Answer: 60 inches The ratio of corresponding linear measurements is 15:9 = 5:3. If the smaller perimeter is 3x, then the larger is 5x. Since 3x + 5x = 96, we have 8x = 96, so x = 12. The larger perimeter is 5(12) = 60 inches. Choice A uses the wrong ratio assignment. Choice B assumes a 4:3 ratio. Choice D uses incorrect arithmetic.
Quadrilateral MNOP is similar to quadrilateral RSTU with a ratio of similarity of 3:7. If MNOP can be mapped to RSTU by a dilation, and the diagonal of MNOP is 21 units, what is the length of the corresponding diagonal of RSTU?
Answer: 49 units When you encounter similar figures with a given ratio of similarity, you're working with proportional relationships between corresponding parts. The key insight is that all corresponding linear measurements (sides, diagonals, perimeters, heights) are related by the same ratio. Here, quadrilateral MNOP is similar to quadrilateral RSTU with a ratio of similarity of 3:7. This means that every linear measurement in MNOP relates to the corresponding measurement in RSTU as 3:7. Since the diagonal of MNOP is 21 units, you can set up a proportion: $\frac{\text{diagonal of MNOP}}{\text{diagonal of RSTU}} = \frac{3}{7}$ Substituting the known value: $\frac{21}{\text{diagonal of RSTU}} = \frac{3}{7}$ Cross-multiplying: $3 \times \text{diagonal of RSTU} = 21 \times 7 = 147$ Therefore: $\text{diagonal of RSTU} = \frac{147}{3} = 49$ units. Choice A (9 units) represents the error of dividing 21 by the scale factor instead of multiplying by the reciprocal ratio. Choice B (35 units) comes from incorrectly using $21 \times \frac{7}{3}$ but making an arithmetic mistake. Choice C (147 units) is the intermediate calculation (21 × 7) before dividing by 3. The correct answer is D) 49 units. Study tip: Always remember that when figure A is similar to figure B with ratio $a:b$ , then corresponding linear measurements follow the same ratio. Set up your proportion carefully, keeping track of which figure corresponds to which part of the ratio.
A trapezoid has parallel sides of lengths 6 units and 10 units, with a height of 4 units. After dilation with scale factor $k$ , the area becomes 128 square units. What is the length of the longer parallel side after dilation?
Answer: 20 units Original area: $\frac{1}{2}(6+10)(4) = 32$ square units. Since the new area is 128, the area scale factor is $\frac{128}{32} = 4 = k^2$ , so $k = 2$ . The longer parallel side becomes $10 \times 2 = 20$ units. Choice A uses the shorter side calculation. Choice C incorrectly uses the area scale factor. Choice D doubles the wrong calculation.
Triangle XYZ undergoes two consecutive dilations: first by scale factor $\frac{2}{3}$ about point A, then by scale factor $\frac{3}{2}$ about point B. If angle Y measures 75° in the original triangle, what is the measure of the corresponding angle in the final image?
Answer: 75° When you encounter problems involving multiple transformations, focus on how each type of transformation affects different properties of geometric figures. Dilations are transformations that resize figures by multiplying all distances from a center point by a scale factor. The key insight here is understanding what dilations preserve versus what they change. Dilations change the size of a figure - they make it larger or smaller depending on whether the scale factor is greater than or less than 1. However, dilations are similarity transformations, meaning they preserve all angle measures, ratios of corresponding sides, and the overall shape of the figure. In this problem, triangle XYZ undergoes two consecutive dilations with scale factors $\frac{2}{3}$ and $\frac{3}{2}$ . Since angle measures are preserved under dilation regardless of the scale factor or center of dilation, the 75° angle Y remains 75° after both transformations. The answer is (A) 75°. The incorrect answers represent common misconceptions: (B) 50° might come from incorrectly multiplying the original angle by $\frac{2}{3}$ ( $75° × \frac{2}{3} = 50°$ ). (C) 100° could result from incorrectly multiplying by $\frac{3}{2}$ ( $75° × \frac{4}{3} ≈ 100°$ ). (D) 112.5° might come from multiplying by $\frac{3}{2}$ ( $75° × \frac{3}{2} = 112.5°$ ). Key takeaway: Dilations always preserve angle measures - only the size changes, never the angles. When you see dilation problems asking about angles, the answer is always the original angle measure.
Two similar polygons have a scale factor of $\frac{4}{7}$ (smaller to larger). If the sum of the perimeters of both polygons is 132 units, what is the perimeter of the smaller polygon?
Answer: 48 units Let the perimeter of the smaller polygon be $p$ . Then the perimeter of the larger polygon is $\frac{7}{4}p$ . The sum is $p + \frac{7}{4}p = \frac{11}{4}p = 132$ . Solving: $p = 132 \cdot \frac{4}{11} = 48$ . Choice B results from incorrectly using $\frac{4}{7}$ as the ratio of larger to smaller. Choices C and D come from setting up the proportion incorrectly or confusing which polygon is which.
Triangle $ABC$ is dilated by a scale factor of $\frac{3}{4}$ to produce triangle $A'B'C'$ . If the perimeter of triangle $ABC$ is 36 units and the area is 54 square units, what is the ratio of the area of triangle $A'B'C'$ to the perimeter of triangle $A'B'C'$ ?
Answer: $\frac{3}{2}$ Under dilation with scale factor $\frac{3}{4}$ , the perimeter scales by $\frac{3}{4}$ and the area scales by $(\frac{3}{4})^2 = \frac{9}{16}$ . New perimeter: $36 \cdot \frac{3}{4} = 27$ . New area: $54 \cdot \frac{9}{16} = \frac{486}{16} = \frac{243}{8}$ . Ratio: $\frac{243/8}{27} = \frac{243}{8} \cdot \frac{1}{27} = \frac{9}{8} \cdot \frac{1}{3} = \frac{3}{2}$ . Choice B uses the area scaling factor incorrectly. Choice C incorrectly uses original values. Choice D confuses the calculation steps.
In the coordinate plane, triangle $ABC$ has vertices at $A(2,4)$ , $B(6,8)$ , and $C(4,2)$ . The triangle is dilated with center at the origin to produce triangle $A'B'C'$ where $A'(3,6)$ . Which statement about the dilation is true?
Answer: The scale factor is $\frac{3}{2}$ and the area increases by factor $\frac{9}{4}$ Since $A(2,4)$ maps to $A'(3,6)$ , the scale factor is $\frac{3}{2}$ (each coordinate is multiplied by $\frac{3}{2}$ ). Under dilation, angles are preserved (not increased), so choice B is wrong. The area scales by the square of the scale factor: $(\frac{3}{2})^2 = \frac{9}{4}$ . Choice A is incomplete about area scaling. Choice D has the wrong scale factor and incorrect area relationship.
Two similar polygons have corresponding sides in the ratio 5:8. If a dilation maps the smaller polygon to the larger polygon, and the difference between their areas is 117 square centimeters, what is the area of the smaller polygon?
Answer: 75 square centimeters The ratio of areas is (8/5)² = 64/25. If the smaller area is A, then the larger area is (64/25)A. The difference is (64/25)A - A = (39/25)A = 117. Solving: A = 117 × (25/39) = 75 square centimeters. Choice A uses the linear ratio incorrectly. Choice C assumes the ratio is 6:8. Choice D uses an incorrect area ratio calculation.
Triangle ABC is dilated by a scale factor of $\frac{3}{4}$ about point P to create triangle A'B'C'. If the perimeter of triangle ABC is 36 units and the area of triangle A'B'C' is 81 square units, what is the area of triangle ABC?
Answer: 144 square units Under a dilation with scale factor k, areas are multiplied by k². Here k = 3/4, so the area ratio is (3/4)² = 9/16. If A'B'C' has area 81, then ABC has area 81 ÷ (9/16) = 81 × (16/9) = 144 square units. Choice A incorrectly uses the linear scale factor. Choice C uses the reciprocal area ratio incorrectly. Choice D uses k³ instead of k².
Rectangle DEFG undergoes a dilation with center at vertex D and scale factor 2.5. If the original rectangle has dimensions 4 cm by 6 cm, and vertex G is originally located at coordinates (4, 6) relative to D as the origin, what are the coordinates of G' after the dilation?
Answer: (10, 15) In a dilation with center D and scale factor 2.5, each coordinate is multiplied by the scale factor. G' = (2.5 × 4, 2.5 × 6) = (10, 15). Choice A incorrectly adds the scale factor. Choice B uses scale factor 2. Choice D uses scale factor 3.

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