Dilations Keep Lines Parallel - Geometry

1. 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?
   Answer: Line $d'$ is parallel to line $d$. 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.
2. 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?
   Answer: No sides will map to coincident lines since the scale factor is not $1$, but all sides will map to parallel lines 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.
3. 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?
   Answer: Lines $\ell_1'$ and $\ell_2'$ are parallel to their respective pre-images, and since $\ell_1 \parallel \ell_2$, we also have $\ell_1' \parallel \ell_2'$ 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.
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 $n'$?
   Answer: $4$ units, because the scale factor $\frac{1}{3}$ reduces the distance between parallel lines by the same factor 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.
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?
   Answer: Both $\ell \parallel \ell'$ and $m$ coincides with $m'$, and the perpendicular relationship between $\ell'$ and $m'$ is preserved 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'.
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?
   Answer: The student is incorrect; the scale factor is $3$ because parallel lines under dilation have proportional constant terms, and $\frac{36}{12} = 3$ 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.
7. 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 $A'B'$?
   Answer: $\frac{3}{4}$, because dilations preserve slopes when the scale factor is positive and the line doesn't pass through the center 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.
8. 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?
   Answer: $3$ lines, because all lines not passing through the center of dilation map to parallel lines regardless of their original relationships 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.
9. 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?
   Answer: 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 $\overline{BC}$ 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.

Start this quiz