derek tried to dilate the isosceles trapezoid abdc with the center of dilation at the origin with a scale factor of 2. check his work and complete the statements. the slope of ac is dropdown. the slope of eg is dropdown. the polygons are not dilations of each other because dropdown with options: the center of dilation is not at (0, 0); the coordinate g should be at (-1, 0); the corresponding sides are not parallel; ac and eg are corresponding sides.

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### Step 1: Find coordinates of points for slope of AC
First, identify coordinates of $ A $ and $ C $. From the graph, $ A(-2, 4) $ and $ C(-3, 0) $? Wait, no, looking at the grid: Let's recheck. Wait, $ C $ is at $ (-3, 0) $? Wait, no, the x-axis: $ C $ is at $ x=-3 $? Wait, no, the grid lines: Let's see, $ A $ is at $ (-2, 4) $? Wait, no, the top base $ AB $: $ A $ is at $ (-2, 4) $? Wait, the y-axis is in the middle. Wait, $ A $ is at $ (-2, 4) $, $ C $ is at $ (-3, 0) $? No, maybe $ A(-2, 4) $, $ C(-3, 0) $? Wait, no, let's check the grid. Wait, the bottom base: $ G $ is at $ (-2, 0) $? Wait, no, the x-axis: $ G $ is at $ (-2, 0) $, $ C $ is at $ (-3, 0) $? Wait, maybe I misread. Let's do it properly.

Wait, the trapezoid $ ABDC $: Let's find coordinates. $ A $: x=-2, y=4? Wait, no, the top side $ AB $ is from x=-2 to x=2, y=4? Wait, the grid: each square is 1 unit. So $ A(-2, 4) $, $ B(2, 4) $, $ C(-3, 0) $, $ D(3, 0) $? Wait, no, $ G $ is at (-2, 0), $ H $ at (2, 0). Wait, maybe $ C $ is at (-3, 0), $ G $ at (-2, 0), $ A $ at (-2, 4), $ E $ at (-1, 2). Wait, maybe I need to get the correct coordinates.

Wait, let's take $ A(-2, 4) $, $ C(-3, 0) $? No, that can't be. Wait, the slope formula is $ m = \frac{y_2 - y_1}{x_2 - x_1} $. Let's find $ A $ and $ C $:

Looking at the graph, $ A $ is at $ (-2, 4) $, $ C $ is at $ (-3, 0) $? No, maybe $ A(-2, 4) $, $ C(-3, 0) $? Wait, no, the distance from $ C $ to $ G $: $ G $ is at (-2, 0), so $ C $ is at (-3, 0), $ G $ at (-2, 0), $ E $ at (-1, 2), $ A $ at (-2, 4). So $ AC $ is from $ A(-2, 4) $ to $ C(-3, 0) $? Wait, no, $ A $ to $ C $: $ A(-2, 4) $, $ C(-3, 0) $? Then slope is $ \frac{0 - 4}{-3 - (-2)} = \frac{-4}{-1} = 4 $? No, that doesn't seem right. Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is wrong. Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is incorrect. Let's check $ E $ and $ G $: $ E(-1, 2) $, $ G(-2, 0) $. Then slope of $ EG $ is $ \frac{0 - 2}{-2 - (-1)} = \frac{-2}{-1} = 2 $. Wait, but let's re-express:

Wait, correct coordinates: Let's assume the grid is 1 unit per square. So:

- $ A(-2, 4) $
- $ B(2, 4) $
- $ C(-3, 0) $ – no, $ C $ is at (-3, 0)? Wait, $ G $ is at (-2, 0), so $ C $ is at (-3, 0), $ G $ at (-2, 0), $ E $ at (-1, 2), $ F $ at (1, 2), $ H $ at (2, 0), $ D $ at (3, 0).

So $ AC $ is from $ A(-2, 4) $ to $ C(-3, 0) $:

Slope of $ AC $: $ m = \frac{y_C - y_A}{x_C - x_A} = \frac{0 - 4}{-3 - (-2)} = \frac{-4}{-1} = 4 $? No, that's not right. Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is wrong. Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is incorrect. Wait, let's check $ E(-1, 2) $ and $ G(-2, 0) $: slope of $ EG $ is $ \frac{0 - 2}{-2 - (-1)} = \frac{-2}{-1} = 2 $.

Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is wrong. Let's re-express:

Wait, the top trapezoid $ ABDC $: $ A(-2, 4) $, $ B(2, 4) $, $ D(3, 0) $, $ C(-3, 0) $. Then $ AC $ is from $ (-2, 4) $ to $ (-3, 0) $: slope $ \frac{0 - 4}{-3 - (-2)} = \frac{-4}{-1} = 4 $. The smaller trapezoid $ EFGH $: $ E(-1, 2) $, $ F(1, 2) $, $ H(2, 0) $, $ G(-2, 0) $. Then $ EG $ is from $ (-1, 2) $ to $ (-2, 0) $: slope $ \frac{0 - 2}{-2 - (-1)} = \frac{-2}{-1} = 2 $.

Now, for dilation with scale factor 2, the slope should remain the same (since dilation preserves parallelism, so slopes of corresponding sides should be equal). But here, slope of $ AC $ is 4, slope of $ EG $ is 2, which are not equal. Wait, but maybe my coordinates are wrong.

Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is incorrect. Let's check the grid again. The x-axis: from -4 to 4. The y-axis: from -4 to 4. $ A $ is at x=-2, y=4; $ B $ at x=2, y=4; $ C $ at x=-3, y=0; $ D $ at x=3, y=0; $ G $ at x=-2, y=0; $ H $ at x=2, y=0; $ E $ at x=-1, y=2; $ F $ at x=1, y=2.

So $ AC $: from (-2, 4) to (-3, 0). So $ \Delta y = 0 - 4 = -4 $, $ \Delta x = -3 - (-2) = -1 $. So slope $ \frac{-4}{-1} = 4 $.

$ EG $: from (-1, 2) to (-2, 0). $ \Delta y = 0 - 2 = -2 $, $ \Delta x = -2 - (-1) = -1 $. So slope $ \frac{-2}{-1} = 2 $.

Now, for dilation with center at origin and scale factor 2, the image of a point $ (x, y) $ is $ (2x, 2y) $. Let's check the coordinates of $ E $ and $ G $. $ E $ is (-1, 2), so its image under dilation (scale 2) should be (-2, 4), which is point $ A $. $ G $ is (-2, 0), so its image under dilation (scale 2) should be (-4, 0), but $ C $ is at (-3, 0), not (-4, 0). Wait, maybe the original trapezoid $ ABDC $ has $ C $ at (-3, 0), $ G $ at (-2, 0). So the corresponding side to $ AC $ should be $ EG $, but the slope of $ AC $ is 4, slope of $ EG $ is 2, which are not equal. Wait, but the options say "the corresponding sides are not parallel" (since parallel sides have equal slopes). So the polygons are not dilations because corresponding sides are not parallel (since their slopes are different, so they are not parallel).

Wait, let's recheck the slope of $ AC $:

Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is wrong. Let's take $ A(-2, 4) $, $ C(-3, 0) $: no, maybe $ A(-2, 4) $, $ C(-3, 0) $ is incorrect. Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is wrong. Let's check $ A(-2, 4) $, $ C(-3, 0) $:

Slope of $ AC $: $ (0 - 4)/(-3 - (-2)) = (-4)/(-1) = 4 $.

Slope of $ EG $: $ (0 - 2)/(-2 - (-1)) = (-2)/(-1) = 2 $.

Since 4 ≠ 2, the slopes are different, so the sides are not parallel. Therefore, the polygons are not dilations because corresponding sides are not parallel.

Now, the first two blanks: slope of $ AC $ is 4? Wait, no, maybe I made a mistake in coordinates. Wait, maybe $ A(-2, 4) $, $ C(-3, 0) $ is wrong. Let's check the grid again. The top base $ AB $ is length 4 (from x=-2 to x=2, so 4 units). The bottom base $ CD $ is length 6 (from x=-3 to x=3, 6 units). The smaller trapezoid $ EFGH $: top base $ EF $ is length 2 (x=-1 to x=1), bottom base $ GH $ is length 4 (x=-2 to x=2). So scale factor from $ EFGH $ to $ ABDC $ should be 2 (since 4/2=2, 6/4=1.5? No, that's inconsistent). Wait, maybe the center of dilation is not origin? No, the problem says center at origin.

Wait, maybe the coordinates are:

$ A(-2, 4) $, $ B(2, 4) $, $ C(-3, 0) $, $ D(3, 0) $

$ E(-1, 2) $, $ F(1, 2) $, $ G(-2, 0) $, $ H(2, 0) $

So $ AC $ is from (-2, 4) to (-3, 0): slope $ (0 - 4)/(-3 - (-2)) = -4 / -1 = 4 $

$ EG $ is from (-1, 2) to (-2, 0): slope $ (0 - 2)/(-2 - (-1)) = -2 / -1 = 2 $

So slope of $ AC $ is 4, slope of $ EG $ is 2.

Now, the third part: the polygons are not dilations because the corresponding sides are not parallel (since their slopes are different, so they are not parallel). The other options: center is at (0,0) (given), so first option is wrong. Coordinate G should be at (-1, 0)? No, because dilation of E(-1,2) is (-2,4)=A, so dilation of G should be C. If G is (-2,0), then dilation (scale 2) is (-4,0), but C is (-3,0), so that's not it. The correct reason is "the corresponding sides are not parallel" because their slopes are different (4 vs 2), so they are not parallel, hence not a dilation (since dilation preserves parallelism).

### Step 1: Slope of AC
Identify coordinates of $ A $ and $ C $: $ A(-2, 4) $, $ C(-3, 0) $.
Slope formula: $ m = \frac{y_2 - y_1}{x_2 - x_1} $.
Substitute: $ m_{AC} = \frac{0 - 4}{-3 - (-2)} = \frac{-4}{-1} = 4 $.

### Step 2: Slope of EG
Identify coordinates of $ E $ and $ G $: $ E(-1, 2) $, $ G(-2, 0) $.
Slope formula: $ m = \frac{y_2 - y_1}{x_2 - x_1} $.
Substitute: $ m_{EG} = \frac{0 - 2}{-2 - (-1)} = \frac{-2}{-1} = 2 $.

### Step 3: Reason for non-dilation
Dilation preserves parallelism (slopes of corresponding sides must be equal). Since $ m_{AC} = 4 $ and $ m_{EG} = 2 $ (unequal), corresponding sides are not parallel.

# Answer:
The slope of $ AC $ is $ \boldsymbol{4} $.
The slope of $ EG $ is $ \boldsymbol{2} $.
The polygons are not dilations of each other because $ \boldsymbol{\text{the corresponding sides are not parallel}} $.

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Step 1: Find coordinates of points for slope of AC

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