is rectangle efgh the result of a dilation of rectangle abcd with a center of dilation at the origin? why or why not?
no, because corresponding sides have different slopes.
yes, because both figures are rectangles and all rectangles are similar.
yes, because corresponding sides are parallel and have lengths in the ratio $\frac{4}{3}$.
no, because the center of dilation is not at $(0, 0)$.

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# Brief Explanations:
To determine if rectangle $ EFGH $ is a dilation of rectangle $ ABCD $ with center at the origin, we analyze dilation properties. Dilation preserves parallelism and scales lengths by a constant ratio.
- First, check parallelism: Sides of rectangles are horizontal/vertical, so corresponding sides (e.g., $ AB \parallel EF $, $ AD \parallel EH $) are parallel.
- Next, check length ratio: Let’s find coordinates. For $ ABCD $, say $ A(-3,3) $, $ B(3,3) $, $ C(3,1) $, $ D(-3,1) $. Length $ AB = 6 $, $ AD = 2 $. For $ EFGH $, $ E(-4,4) $, $ F(4,4) $, $ G(4,1) $, $ H(-4,1) $. Length $ EF = 8 $, $ EH = 3 $? Wait, no—wait, $ AD $ is from $ y=1 $ to $ y=3 $, so length $ 2 $; $ EH $ is from $ y=1 $ to $ y=4 $, length $ 3 $? Wait, no, maybe better: $ AB $ length: $ ABCD $: from $ x=-3 $ to $ x=3 $, so $ 6 $; $ EFGH $: from $ x=-4 $ to $ x=4 $, so $ 8 $. Ratio $ \frac{8}{6} = \frac{4}{3} $. $ AD $: $ ABCD $ from $ y=1 $ to $ y=3 $, length $ 2 $; $ EH $: $ EFGH $ from $ y=1 $ to $ y=4 $, length $ 3 $. Ratio $ \frac{3}{2} $? Wait, no, maybe I messed up coordinates. Wait, the grid: $ D $ is at $ (-3,0) $? Wait, the x-axis is at $ y=0 $. So $ D(-3,0) $, $ C(3,0) $, $ B(3,3) $, $ A(-3,3) $? Wait, the blue rectangle $ ABCD $: let's re-express. Let's take $ A(-3,3) $, $ B(3,3) $, $ C(3,1) $, $ D(-3,1) $. Then $ E(-4,4) $, $ F(4,4) $, $ G(4,1) $, $ H(-4,1) $. So $ AB $ length: $ 3 - (-3) = 6 $; $ EF $ length: $ 4 - (-4) = 8 $. Ratio $ \frac{EF}{AB} = \frac{8}{6} = \frac{4}{3} $. $ AD $ length: $ 3 - 1 = 2 $; $ EH $ length: $ 4 - 1 = 3 $. Wait, ratio $ \frac{EH}{AD} = \frac{3}{2} $? No, that can't be. Wait, maybe $ ABCD $ has $ A(-3,3) $, $ B(3,3) $, $ C(3,0) $, $ D(-3,0) $? Then $ AD $ length is $ 3 - 0 = 3 $, $ AB $ length $ 6 $. $ EFGH $: $ E(-4,4) $, $ F(4,4) $, $ G(4,0) $, $ H(-4,0) $. Then $ EF = 8 $, $ EH = 4 $. Ratio $ \frac{EF}{AB} = \frac{8}{6} = \frac{4}{3} $, $ EH/AD = \frac{4}{3} $. Ah, that makes sense. So corresponding sides are parallel (horizontal/vertical, so slopes 0 or undefined, parallel) and lengths are in ratio $ \frac{4}{3} $. Dilation from origin: scaling factor $ \frac{4}{3} $ would map $ ABCD $ to $ EFGH $ if center is origin. Wait, but the last option says "No, because the center of dilation is not at (0,0)"—but the question says center is origin. Wait, no, the options: let's check each:

1. "No, because corresponding sides have different slopes": Sides of rectangles are horizontal/vertical, so slopes 0 (horizontal) or undefined (vertical), so corresponding sides have same slopes (parallel). So this is wrong.

2. "Yes, because both figures are rectangles and all rectangles are similar": Not all rectangles are similar (e.g., 2x3 and 4x5 rectangles are not similar). Similarity requires proportional sides. So this is wrong.

3. "Yes, because corresponding sides are parallel and have lengths in the ratio $ \frac{4}{3} $": As we saw, if $ ABCD $ has sides 6 and 3 (wait, no, if $ AB = 6 $, $ AD = 3 $, then $ EFGH $ has $ EF = 8 $, $ EH = 4 $? Wait, no, earlier mistake: if $ D $ is at $ (-3,0) $, $ A $ at $ (-3,3) $, so $ AD $ length 3; $ E $ at $ (-4,4) $, $ H $ at $ (-4,1) $, so $ EH $ length 3? No, $ y $-coordinate of $ E $ is 4, $ H $ is 1, so length 3? Wait, no, $ E(-4,4) $, $ H(-4,1) $: vertical length is $ 4 - 1 = 3 $. $ A(-3,3) $, $ D(-3,0) $: vertical length $ 3 - 0 = 3 $. Wait, that's same? No, $ AB $: $ A(-3,3) $, $ B(3,3) $: horizontal length $ 6 $. $ EF $: $ E(-4,4) $, $ F(4,4) $: horizontal length $ 8 $. So ratio $ 8/6 = 4/3 $. $ AD $: $ 3 $, $ EH $: $ 3 $? No, that can't be. Wait, maybe $ ABCD $ is from $ x=-3 $ to $ x=3 $ (length 6) and $ y=1 $ to $ y=3 $ (length 2), so $ AB=6 $, $ AD=2 $. $ EFGH $ from $ x=-4 $ to $ x=4 $ (length 8) and $ y=1 $ to $ y=4 $ (length 3). Then ratio $ 8/6 = 4/3 $, $ 3/2 $? No, that's inconsistent. Wait, maybe the rectangles are $ ABCD $: $ A(-3,3) $, $ B(3,3) $, $ C(3,1) $, $ D(-3,1) $ (so $ AB=6 $, $ AD=2 $) and $ EFGH $: $ E(-4,4) $, $ F(4,4) $, $ G(4,1) $, $ H(-4,1) $ (so $ EF=8 $, $ EH=3 $). Wait, that's not proportional. Wait, maybe I misread the grid. Alternatively, the key is: Dilation from origin means that the line from origin to each vertex of $ ABCD $ should pass through the corresponding vertex of $ EFGH $, and the ratio of distances from origin should be constant. Let's check coordinates:

- $ A $: let's say $ A(-3,3) $, distance from origin: $ \sqrt{(-3)^2 + 3^2} = \sqrt{18} = 3\sqrt{2} $.
- $ E $: $ (-4,4) $, distance: $ \sqrt{(-4)^2 + 4^2} = \sqrt{32} = 4\sqrt{2} $. Ratio $ \frac{4\sqrt{2}}{3\sqrt{2}} = \frac{4}{3} $.
- $ B(3,3) $: distance $ 3\sqrt{2} $, $ F(4,4) $: $ 4\sqrt{2} $, ratio $ 4/3 $.
- $ C(3,1) $: distance $ \sqrt{9 + 1} = \sqrt{10} $, $ G(4,1) $: $ \sqrt{16 + 1} = \sqrt{17} $. Wait, that's not $ 4/3 $ of $ \sqrt{10} $. Wait, no, maybe $ C $ is $ (3,0) $, $ G(4,0) $. Then $ C(3,0) $, distance $ 3 $; $ G(4,0) $, distance $ 4 $, ratio $ 4/3 $. $ D(-3,0) $, distance $ 3 $; $ H(-4,0) $, distance $ 4 $, ratio $ 4/3 $. Ah, that makes sense. So $ ABCD $ has vertices $ A(-3,3) $, $ B(3,3) $, $ C(3,0) $, $ D(-3,0) $. Then $ E(-4,4) $, $ F(4,4) $, $ G(4,0) $, $ H(-4,0) $. Now, check ratios:

- $ AB $: length $ 6 $ (from $ -3 $ to $ 3 $ on x-axis), $ EF $: length $ 8 $ (from $ -4 $ to $ 4 $), ratio $ 8/6 = 4/3 $.
- $ AD $: length $ 3 $ (from $ 0 $ to $ 3 $ on y-axis), $ EH $: length $ 4 $ (from $ 0 $ to $ 4 $ on y-axis), ratio $ 4/3 $.
- Lines from origin: $ OA $ is from $ (0,0) $ to $ (-3,3) $, slope $ -1 $; $ OE $ is from $ (0,0) $ to $ (-4,4) $, slope $ -1 $, so same line (collinear). Similarly, $ OB $ and $ OF $ (slope $ 1 $), $ OC $ and $ OG $ (slope $ 0 $), $ OD $ and $ OH $ (slope $ 0 $ in negative x). So vertices are collinear with origin, and distances are in ratio $ 4/3 $. Thus, dilation with center origin and scale factor $ 4/3 $ maps $ ABCD $ to $ EFGH $.

Now, check options:

- First option: "No, because corresponding sides have different slopes"—incorrect, sides are parallel (same slopes: horizontal/vertical).
- Second option: "Yes, because both figures are rectangles and all rectangles are similar"—incorrect, not all rectangles are similar (needs proportional sides).
- Third option: "Yes, because corresponding sides are parallel and have lengths in the ratio $ \frac{4}{3} $"—correct, as we saw sides are parallel (horizontal/vertical) and lengths scale by $ 4/3 $.
- Fourth option: "No, because the center of dilation is not at (0, 0)"—incorrect, the question states center is origin, and our analysis shows it is a dilation from origin.

# Answer:
Yes, because corresponding sides are parallel and have lengths in the ratio $\frac{4}{3}$ (the option with this explanation).

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