Dilations Keep Lines Parallel
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Geometry › Dilations Keep Lines Parallel
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, but $$\ell_1'$$ and $$\ell_2'$$ intersect even though $$\ell_1 \parallel \ell_2$$
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, and since $$\ell_1 \parallel \ell_2$$, we also have $$\ell_1' \parallel \ell_2'$$
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.
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 $$A'B'$$?
$$\frac{9}{4}$$, because the slope gets multiplied by the scale factor when the line is mapped to a parallel line under dilation
$$\frac{4}{3}$$, because dilations with scale factor $$3$$ transform slopes by taking their reciprocal and multiplying by the scale factor
$$-\frac{3}{4}$$, because dilations reverse the orientation of lines that do not pass through the center of dilation
$$\frac{3}{4}$$, because dilations preserve slopes when the scale factor is positive and the line doesn't pass through the center
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.
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?
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 $$m'$$, 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
Both $$\ell \parallel \ell'$$ and $$m$$ coincides with $$m'$$, and the perpendicular relationship between $$\ell'$$ and $$m'$$ 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'.
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 $d'$ intersects line $d$ at $O$.
Triangle $JKL$ is congruent to triangle $J'K'L'$.
Line $d'$ is parallel to line $d$.
Line $c'$ is perpendicular to line $c$.
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.