Geometry Quiz: Dilations Keep Lines Parallel

9 questions · exam conditions

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Dilations Keep Lines ParallelQuestion 1 of 9

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

Lines

\ell_1'

and

\ell_2'

are parallel to each other and perpendicular to their respective pre-images, demonstrating that dilations can change angle relationships

Lines

\ell_1'

and

\ell_2'

are parallel to their respective pre-images, and since

\ell_1 \parallel \ell_2

, we also have

\ell_1' \parallel \ell_2'

Lines

\ell_1'

and

\ell_2'

intersect at the origin, while their pre-images

\ell_1

and

\ell_2

intersect at a different point

Lines

\ell_1'

and

\ell_2'

are parallel to their respective pre-images, but

\ell_1'

and

\ell_2'

intersect even though

\ell_1 \parallel \ell_2

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Geometry Quiz

Geometry Quiz: Dilations Keep Lines Parallel

Practice Dilations Keep Lines Parallel in Geometry 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 Dilations Keep Lines Parallel, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

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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.

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Question 1

Lines $\ell_1'$ and $\ell_2'$ are parallel to each other and perpendicular to their respective pre-images, demonstrating that dilations can change angle relationships
Lines $\ell_1'$ and $\ell_2'$ are parallel to their respective pre-images, and since $\ell_1 \parallel \ell_2$ , we also have $\ell_1' \parallel \ell_2'$ (correct answer)
Lines $\ell_1'$ and $\ell_2'$ intersect at the origin, while their pre-images $\ell_1$ and $\ell_2$ intersect at a different point
Lines $\ell_1'$ and $\ell_2'$ are parallel to their respective pre-images, but $\ell_1'$ and $\ell_2'$ intersect even though $\ell_1 \parallel \ell_2$

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 $C$ maps point $A$ to point $A'$ and point $B$ to point $B'$ . Line $AB$ does not pass through $C$ . If $\overrightarrow{CA} = (3, -2)$ and $\overrightarrow{CA'} = (9, -6)$ , and line $AB$ has slope $\frac{3}{4}$ , what is the slope of line $A'B'$ ?

$\frac{3}{4}$ , because dilations preserve slopes when the scale factor is positive and the line doesn't pass through the center (correct answer)
$\frac{4}{3}$ , because dilations with scale factor $3$ transform slopes by taking their reciprocal and multiplying by the scale factor
$\frac{9}{4}$ , because the slope gets multiplied by the scale factor when the line is mapped to a parallel line under dilation
$-\frac{3}{4}$ , 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 $a$ with equation $y = 2x + 1$ , line $b$ with equation $y = 2x + 5$ , and line $c$ with equation $y = -x + 3$ . All three lines are dilated with center $(-1, 3)$ and scale factor $2$ . How many of the image lines will be parallel to their corresponding original lines?

$0$ lines, because dilations centered away from the origin change the slopes of all lines by the scale factor
$1$ line, because only the line that passes through the center of dilation will remain parallel to itself
$2$ lines, because lines $a$ and $b$ are parallel to each other and both avoid the center of dilation
$3$ lines, because all lines not passing through the center of dilation map to parallel lines regardless of their original relationships (correct answer)

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 $m$ and $n$ are each $6$ units away from point $O$ , with $O$ located between the two lines. A dilation with center $O$ and scale factor $\frac{1}{3}$ is applied. After the dilation, what is the distance between the image lines $m'$ and $n'$ ?

$4$ units, because the scale factor $\frac{1}{3}$ reduces the distance between parallel lines by the same factor (correct answer)
$6$ units, because dilations preserve distances between parallel lines when the center lies between them
$12$ units, because the images of the lines move outward from the center, increasing their separation
$2$ units, because each line moves to $\frac{1}{3}$ of its original distance from $O$ , making them $2$ 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 $O$ and scale factor $3$ maps line $\ell$ to line $\ell'$ . Line $m$ is perpendicular to line $\ell$ and passes through point $O$ . After the same dilation, which statement about the relationship between the original lines and their images is true?

Both $\ell \parallel \ell'$ and $m$ coincides with $m'$ , demonstrating that dilations preserve angle relationships between all lines
Both $\ell \parallel \ell'$ and $m$ coincides with $m'$ , but the perpendicular relationship between $\ell'$ and $m'$ is not preserved
Line $\ell$ coincides with $\ell'$ and $m \parallel m'$ , since dilations preserve perpendicular relationships by keeping both lines unchanged
Both $\ell \parallel \ell'$ and $m$ coincides with $m'$ , and the perpendicular relationship between $\ell'$ and $m'$ is preserved (correct answer)

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 $\ell$ has equation $3x + 4y = 12$ and does not pass through the origin. After a dilation with center at the origin and scale factor $k > 0$ , the image line $\ell'$ has equation $3x + 4y = 36$ . A student claims that since the coefficients of $x$ and $y$ are unchanged, the dilation must have scale factor $1$ . Which statement correctly evaluates this reasoning?

The student is correct; unchanged coefficients of $x$ and $y$ always indicate a scale factor of $1$ in dilations centered at origin
The student is incorrect; the scale factor is $3$ because the constant term changed from $12$ to $36$ , and $36 = 3 \times 12$
The student is incorrect; the scale factor is $3$ because parallel lines under dilation have proportional constant terms, and $\frac{36}{12} = 3$ (correct answer)
The student is incorrect; while the lines are parallel as expected, the scale factor is $\frac{1}{3}$ based on the ratio $\frac{12}{36}$

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

Consider the figure where quadrilateral $ABCD$ undergoes a dilation with center $P$ and scale factor $\frac{1}{2}$ . Point $P$ lies on side $\overline{BC}$ but not at vertices $B$ or $C$ . Which statement correctly describes what happens to the sides of the quadrilateral?

Sides $\overline{AB}$ , $\overline{CD}$ , and $\overline{AD}$ each map to parallel segments, while side $\overline{BC}$ maps to a segment contained within the original $\overline{BC}$ (correct answer)
All four sides map to parallel segments of equal length, since dilations preserve parallelism and the scale factor applies uniformly to all sides
Sides $\overline{AB}$ , $\overline{CD}$ , and $\overline{AD}$ each map to parallel segments, while side $\overline{BC}$ remains completely unchanged in position and length
Only sides $\overline{AB}$ and $\overline{CD}$ map to parallel segments, while sides $\overline{BC}$ and $\overline{AD}$ 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 8

Refer to the diagram. Rectangle $PQRS$ is dilated with center $T$ and scale factor $\frac{3}{2}$ to produce rectangle $P'Q'R'S'$ . Point $T$ is located at the intersection of diagonals $\overline{PR}$ and $\overline{QS}$ . Which sides of the original rectangle will be mapped to lines that coincide with the original sides?

All four sides $\overline{PQ}$ , $\overline{QR}$ , $\overline{RS}$ , and $\overline{SP}$ will map to coincident lines since $T$ is the center of the rectangle
Only the diagonals $\overline{PR}$ and $\overline{QS}$ will map to coincident lines, while all sides map to parallel lines
No sides will map to coincident lines since the scale factor is not $1$ , but all sides will map to parallel lines (correct answer)
Opposite sides $\overline{PQ}$ with $\overline{RS}$ and $\overline{QR}$ with $\overline{SP}$ 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.

Question 9

A dilation centered at $O$ maps triangle $JKL$ to triangle $J'K'L'$ . Line $c$ passes through $O$ , and line $d$ does not pass through $O$ . Which statement must be true after the dilation?

Line $d'$ is parallel to line $d$ . (correct answer)
Line $d'$ intersects line $d$ at $O$ .
Line $c'$ is perpendicular to line $c$ .
Triangle $JKL$ is congruent to triangle $J'K'L'$ .

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.