Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Opening subject page...

Loading your content

Geometry

Geometry Help: Dilations Keep Lines Parallel

Review real example questions for Dilations Keep Lines Parallel in Geometry.

Question 1

In a coordinate plane, line ℓ1\ell_1ℓ1​ passes through points (1,4)(1, 4)(1,4) and (5,8)(5, 8)(5,8), while line ℓ2\ell_2ℓ2​ passes through points (2,3)(2, 3)(2,3) and (6,7)(6, 7)(6,7). A dilation with center (0,0)(0, 0)(0,0) and scale factor 222 is applied to both lines. Which statement about the images ℓ1′\ell_1'ℓ1′​ and ℓ2′\ell_2'ℓ2′​ is correct?

  1. Lines ℓ1′\ell_1'ℓ1′​ and ℓ2′\ell_2'ℓ2′​ are parallel to each other and perpendicular to their respective pre-images, demonstrating that dilations can change angle relationships
  2. Lines ℓ1′\ell_1'ℓ1′​ and ℓ2′\ell_2'ℓ2′​ are parallel to their respective pre-images, and since ℓ1∥ℓ2\ell_1 \parallel \ell_2ℓ1​∥ℓ2​, we also have ℓ1′∥ℓ2′\ell_1' \parallel \ell_2'ℓ1′​∥ℓ2′​
  3. Lines ℓ1′\ell_1'ℓ1′​ and ℓ2′\ell_2'ℓ2′​ intersect at the origin, while their pre-images ℓ1\ell_1ℓ1​ and ℓ2\ell_2ℓ2​ intersect at a different point
  4. Lines ℓ1′\ell_1'ℓ1′​ and ℓ2′\ell_2'ℓ2′​ are parallel to their respective pre-images, but ℓ1′\ell_1'ℓ1′​ and ℓ2′\ell_2'ℓ2′​ intersect even though ℓ1∥ℓ2\ell_1 \parallel \ell_2ℓ1​∥ℓ2​
Explanation: First, let's find the slopes: ℓ₁ has slope (8-4)/(5-1) = 1, and ℓ₂ has slope (7-3)/(6-2) = 1. Since both lines have slope 1, ℓ₁ ∥ ℓ₂. Since neither line passes through the center (0,0), both map to parallel lines under dilation. The image lines ℓ₁' and ℓ₂' each have slope 1 (same as their pre-images) and are therefore parallel to their respective pre-images. Since dilations preserve parallelism between lines, ℓ₁' ∥ ℓ₂'. Choice A incorrectly states that images are perpendicular to pre-images. Choice C incorrectly describes intersection points. Choice D incorrectly states that parallel lines become intersecting lines.

Question 2

A dilation with center CCC maps point AAA to point A′A'A′ and point BBB to point B′B'B′. Line ABABAB does not pass through CCC. If CA→=(3,−2)\overrightarrow{CA} = (3, -2)CA=(3,−2) and CA′→=(9,−6)\overrightarrow{CA'} = (9, -6)CA′=(9,−6), and line ABABAB has slope 34\frac{3}{4}43​, what is the slope of line A′B′A'B'A′B′?

  1. 34\frac{3}{4}43​, because dilations preserve slopes when the scale factor is positive and the line doesn't pass through the center
  2. 43\frac{4}{3}34​, because dilations with scale factor 333 transform slopes by taking their reciprocal and multiplying by the scale factor
  3. 94\frac{9}{4}49​, because the slope gets multiplied by the scale factor when the line is mapped to a parallel line under dilation
  4. −34-\frac{3}{4}−43​, because dilations reverse the orientation of lines that do not pass through the center of dilation
Explanation: From the given vectors, the scale factor is |CA'|/|CA| = 9/3 = 3 (or we can see that CA' = 3·CA). Since line AB does not pass through center C, dilation maps it to a parallel line A'B'. Parallel lines have equal slopes, and dilations preserve slopes regardless of the scale factor. Therefore, line A'B' has slope 3/4. Choice B incorrectly suggests slope inversion. Choice C incorrectly multiplies slope by scale factor. Choice D incorrectly suggests orientation reversal.

Question 3

Consider three lines: line aaa with equation y=2x+1y = 2x + 1y=2x+1, line bbb with equation y=2x+5y = 2x + 5y=2x+5, and line ccc with equation y=−x+3y = -x + 3y=−x+3. All three lines are dilated with center (−1,3)(-1, 3)(−1,3) and scale factor 222. How many of the image lines will be parallel to their corresponding original lines?

  1. 000 lines, because dilations centered away from the origin change the slopes of all lines by the scale factor
  2. 111 line, because only the line that passes through the center of dilation will remain parallel to itself
  3. 222 lines, because lines aaa and bbb are parallel to each other and both avoid the center of dilation
  4. 333 lines, because all lines not passing through the center of dilation map to parallel lines regardless of their original relationships
Explanation: We need to check which lines pass through the center (-1, 3). For line a: 3 = 2(-1) + 1 = -1 ≠ 3, so line a doesn't pass through the center. For line b: 3 = 2(-1) + 5 = 3 ✓, so line b passes through the center. For line c: 3 = -(-1) + 3 = 4 ≠ 3, so line c doesn't pass through the center. Lines a and c don't pass through the center, so they map to parallel lines. Line b passes through the center, so it maps to itself (which is trivially parallel to the original). Therefore, all 3 image lines are parallel to their corresponding originals. Choice A incorrectly suggests slopes change. Choice B incorrectly counts only the line through the center. Choice C undercounts by excluding line c.

Question 4

Two parallel lines mmm and nnn are each 666 units away from point OOO, with OOO located between the two lines. A dilation with center OOO and scale factor 13\frac{1}{3}31​ is applied. After the dilation, what is the distance between the image lines m′m'm′ and n′n'n′?

  1. 444 units, because the scale factor 13\frac{1}{3}31​ reduces the distance between parallel lines by the same factor
  2. 666 units, because dilations preserve distances between parallel lines when the center lies between them
  3. 121212 units, because the images of the lines move outward from the center, increasing their separation
  4. 222 units, because each line moves to 13\frac{1}{3}31​ of its original distance from OOO, making them 222 units apart
Explanation: Since O lies between the parallel lines m and n, and each line is 6 units from O, the original distance between the lines is 6 + 6 = 12 units. Under dilation with center O and scale factor 1/3, each line maps to a parallel line at distance (1/3) × 6 = 2 units from O. Since the lines are on opposite sides of O, the distance between the image lines is 2 + 2 = 4 units. The scale factor 1/3 reduces the distance between parallel lines by the same factor: 12 × (1/3) = 4. Choice B incorrectly suggests distances are preserved. Choice C incorrectly suggests increased separation. Choice D incorrectly calculates the final distance.

Question 5

A dilation with center OOO and scale factor 333 maps line ℓ\ellℓ to line ℓ′\ell'ℓ′. Line mmm is perpendicular to line ℓ\ellℓ and passes through point OOO. After the same dilation, which statement about the relationship between the original lines and their images is true?

  1. Both ℓ∥ℓ′\ell \parallel \ell'ℓ∥ℓ′ and mmm coincides with m′m'm′, demonstrating that dilations preserve angle relationships between all lines
  2. Both ℓ∥ℓ′\ell \parallel \ell'ℓ∥ℓ′ and mmm coincides with m′m'm′, but the perpendicular relationship between ℓ′\ell'ℓ′ and m′m'm′ is not preserved
  3. Line ℓ\ellℓ coincides with ℓ′\ell'ℓ′ and m∥m′m \parallel m'm∥m′, since dilations preserve perpendicular relationships by keeping both lines unchanged
  4. Both ℓ∥ℓ′\ell \parallel \ell'ℓ∥ℓ′ and mmm coincides with m′m'm′, and the perpendicular relationship between ℓ′\ell'ℓ′ and m′m'm′ is preserved
Explanation: Since line ℓ does not pass through center O, the dilation maps it to a parallel line ℓ', so ℓ ∥ ℓ'. Since line m passes through center O, it remains unchanged under dilation, so m coincides with m'. Dilations preserve angle measures, so the 90° angle between ℓ and m is preserved between ℓ' and m' (which is the same as m). Choice A incorrectly suggests dilations preserve angle relationships between ALL lines. Choice B incorrectly states that perpendicularity is not preserved. Choice C incorrectly states that ℓ coincides with ℓ' and that m ∥ m'.

Question 6

Line ℓ\ellℓ has equation 3x+4y=123x + 4y = 123x+4y=12 and does not pass through the origin. After a dilation with center at the origin and scale factor k>0k > 0k>0, the image line ℓ′\ell'ℓ′ has equation 3x+4y=363x + 4y = 363x+4y=36. A student claims that since the coefficients of xxx and yyy are unchanged, the dilation must have scale factor 111. Which statement correctly evaluates this reasoning?

  1. The student is correct; unchanged coefficients of xxx and yyy always indicate a scale factor of 111 in dilations centered at origin
  2. The student is incorrect; the scale factor is 333 because the constant term changed from 121212 to 363636, and 36=3×1236 = 3 \times 1236=3×12
  3. The student is incorrect; the scale factor is 333 because parallel lines under dilation have proportional constant terms, and 3612=3\frac{36}{12} = 31236​=3
  4. The student is incorrect; while the lines are parallel as expected, the scale factor is 13\frac{1}{3}31​ based on the ratio 1236\frac{12}{36}3612​
Explanation: When a line not passing through the center is dilated, it maps to a parallel line. Parallel lines have the same coefficients for x and y terms but different constant terms. For a dilation with scale factor k, if the original line is ax + by = c, the image line is ax + by = kc. Here, 3x + 4y = 12 maps to 3x + 4y = 36, so k = 36/12 = 3. The student's reasoning is flawed because unchanged coefficients indicate parallelism, not scale factor 1. Choice A incorrectly supports the student's flawed reasoning. Choice B gives the correct scale factor but with incomplete reasoning. Choice D incorrectly inverts the ratio.

Question 7

A dilation centered at OOO maps triangle JKLJKLJKL to triangle J′K′L′J'K'L'J′K′L′. Line ccc passes through OOO, and line ddd does not pass through OOO. Which statement must be true after the dilation?

  1. Line d′d'd′ is parallel to line ddd.
  2. Line d′d'd′ intersects line ddd at OOO.
  3. Line c′c'c′ is perpendicular to line ccc.
  4. Triangle JKLJKLJKL is congruent to triangle J′K′L′J'K'L'J′K′L′.
Explanation: This problem tests understanding of dilation effects on lines with different positions relative to the center. The fundamental rule is that lines passing through the center of dilation map onto themselves, while lines not passing through the center map to parallel lines. With O as the center, line c passes through O, and line d does not pass through O. Therefore, line c maps onto itself (c' = c), and line d maps to a line parallel to d (d' is parallel to d). The correct answer is A. Lines never become perpendicular under dilation, and triangles are similar but not congruent unless the scale factor is 1. A common error is thinking all lines behave the same way, but the key distinction is whether they pass through the center. Always classify lines by their relationship to the center before determining their images.

Question 8

Consider the figure where quadrilateral ABCDABCDABCD undergoes a dilation with center PPP and scale factor 12\frac{1}{2}21​. Point PPP lies on side BC‾\overline{BC}BC but not at vertices BBB or CCC. Which statement correctly describes what happens to the sides of the quadrilateral?

  1. Sides AB‾\overline{AB}AB, CD‾\overline{CD}CD, and AD‾\overline{AD}AD each map to parallel segments, while side BC‾\overline{BC}BC maps to a segment contained within the original BC‾\overline{BC}BC
  2. All four sides map to parallel segments of equal length, since dilations preserve parallelism and the scale factor applies uniformly to all sides
  3. Sides AB‾\overline{AB}AB, CD‾\overline{CD}CD, and AD‾\overline{AD}AD each map to parallel segments, while side BC‾\overline{BC}BC remains completely unchanged in position and length
  4. Only sides AB‾\overline{AB}AB and CD‾\overline{CD}CD map to parallel segments, while sides BC‾\overline{BC}BC and AD‾\overline{AD}AD remain unchanged since they connect to the center
Explanation: Since P lies on side BC, the line containing BC passes through the center of dilation. Therefore, this line remains unchanged, but the segment BC itself gets scaled to a shorter segment contained within the original BC. The other three sides (AB, CD, AD) do not lie on lines through P, so they map to parallel segments. Choice B is wrong because BC doesn't map to a parallel segment of the same length. Choice C is wrong because BC changes length even though its containing line is unchanged. Choice D incorrectly describes which sides are affected.

Question 9

Refer to the diagram. Rectangle PQRSPQRSPQRS is dilated with center TTT and scale factor 32\frac{3}{2}23​ to produce rectangle P′Q′R′S′P'Q'R'S'P′Q′R′S′. Point TTT is located at the intersection of diagonals PR‾\overline{PR}PR and QS‾\overline{QS}QS​. Which sides of the original rectangle will be mapped to lines that coincide with the original sides?

  1. All four sides PQ‾\overline{PQ}PQ​, QR‾\overline{QR}QR​, RS‾\overline{RS}RS, and SP‾\overline{SP}SP will map to coincident lines since TTT is the center of the rectangle
  2. Only the diagonals PR‾\overline{PR}PR and QS‾\overline{QS}QS​ will map to coincident lines, while all sides map to parallel lines
  3. No sides will map to coincident lines since the scale factor is not 111, but all sides will map to parallel lines
  4. Opposite sides PQ‾\overline{PQ}PQ​ with RS‾\overline{RS}RS and QR‾\overline{QR}QR​ with SP‾\overline{SP}SP will map to the same coincident lines respectively
Explanation: Point T is the center of rectangle PQRS (intersection of diagonals). However, none of the sides of the rectangle pass through point T - the sides are the edges of the rectangle, while T is at the interior center. Since no side passes through the center of dilation T, all sides map to parallel lines (not coincident lines). The segments themselves will be longer due to the scale factor 3/2 > 1. Choice A incorrectly assumes sides pass through the center. Choice B correctly identifies that diagonals pass through T and remain on the same lines, but incorrectly suggests some sides might be coincident. Choice D incorrectly describes the mapping of opposite sides.