Question

analyzing a dilation
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 4.
the slope of eg is 2.
the polygons are not dilations of each other because
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

Explanation:

Step1: Recall Dilation Properties

Dilation preserves slope (parallelism) and scales distances from the center. Center is origin, scale factor 2.

Step2: Analyze Coordinates

Point C: (-3, 0), A: (-2, 4). For dilation, G (image of C) should be \( (-3\times2, 0\times2)=(-6,0) \)? Wait, no—wait, original trapezoid ABDC: let's find coordinates. Wait, maybe I misread. Wait, the small trapezoid EFGH: G is at (-2,0)? Wait, no, the grid: C is at (-3,0)? Wait, no, the x-axis: C is at (-3,0)? Wait, the leftmost point C: x=-3, y=0. A is at (-2,4)? Wait, no, the blue trapezoid: A is at (-2,4)? Wait, no, the top base AB: A(-2,4), B(2,4)? Wait, no, the grid: each square is 1 unit. So A is (-2,4), B(2,4), C(-3,0), D(3,0). Then EFGH: E(-1,2), F(1,2), G(-2,0), H(2,0). Now, dilation with center origin, scale factor 2: image of G(-2,0) should be (-4,0), but original C is (-3,0). Wait, no—wait, the problem is Derek's dilation. Wait, the question is why they are not dilations. Let's check the options. Option: "the coordinate G should be at (-1, 0)"? No, wait, maybe I messed up. Wait, original trapezoid ABDC: let's find coordinates of C and A. C is at (-3,0), A is at (-2,4). Slope of AC: \( \frac{4 - 0}{-2 - (-3)} = \frac{4}{1}=4 \), correct. Slope of EG: E is (-1,2), G is (-2,0). Slope: \( \frac{2 - 0}{-1 - (-2)} = \frac{2}{1}=2 \). Wait, but for dilation, corresponding sides should be parallel (same slope), but AC has slope 4, EG has slope 2—wait, no, that can't be. Wait, no, maybe corresponding sides are AC and EG? Wait, no, dilation should map AC to EG with scale factor 2. Let's check coordinates. If center is origin, scale factor 2, then vector from origin to C is (-3,0), so image should be (-6,0), but G is (-2,0). Wait, no, maybe the original trapezoid is EFGH, and Derek tried to dilate to ABDC? Wait, no, the problem says "Derek tried to dilate the isosceles trapezoid ABDC with center at origin, scale factor 2" to get EFGH? Wait, no, the blue trapezoid is ABDC, the smaller is EFGH. So dilation of ABDC (larger) by scale factor 1/2 should get EFGH. But G is at (-2,0), while C is at (-3,0). Wait, the option: "the coordinate G should be at (-1, 0)"? No, wait, maybe I made a mistake. Wait, let's re-express. Let's find coordinates:

ABDC: A(-2,4), B(2,4), C(-3,0), D(3,0).

EFGH: E(-1,2), F(1,2), G(-2,0), H(2,0).

Dilation center origin, scale factor 2: to get ABDC from EFGH, scale factor 2. So G(-2,0) scaled by 2 is (-4,0), but C is (-3,0). Not matching. Wait, the option: "the coordinate G should be at (-1, 0)"? No, that's not. Wait, another option: "the corresponding sides are not parallel"—but AC slope 4, EG slope 2: not parallel. Wait, but the first two slopes are 4 and 2, which are different. Wait, but the problem's first two dropdowns have slope of AC as 4 (correct) and EG as 2 (correct). Then the reason: let's check the options. The options are:

- the center of dilation is not at (0,0) → no, center is origin.

- the coordinate G should be at (-1, 0) → no.

- the corresponding sides are not parallel → AC slope 4, EG slope 2: not parallel. Wait, but the fourth option is "AC and EG are corresponding sides"—no, that's a statement, not a reason. Wait, no, the options in the dropdown: let's recheck. The options are:

1. the center of dilation is not at (0,0)

2. the coordinate G should be at (-1, 0)

3. the corresponding sides are not parallel

4. AC and EG are corresponding sides

Wait, the correct reason: for dilation, corresponding sides must be parallel (same slope). AC has slope 4, EG has slope 2: not parallel. So the polygons are not dilations because corresponding sides are not parallel. Wait, but the first two slopes are 4 and 2, which are different. So the answer is "the corresponding sides are not parallel". Wait, but let's check again. Wait, maybe I misidentified corresponding sides. Maybe AC and EG are not corresponding sides? No, in dilation, the sides should correspond. Alternatively, maybe the scale factor is wrong. Wait, no, the problem says "the polygons are not dilations of each other because..." Let's check the coordinates again.

Wait, maybe the error is in the coordinates of G. Let's see: if the center is origin, scale factor 2, then to get EFGH from ABDC, scale factor 1/2. So C(-3,0) scaled by 1/2 is (-1.5,0), but G is (-2,0). Not matching. But the option "the coordinate G should be at (-1, 0)" is not correct. Wait, maybe I made a mistake in coordinates. Let's look at the grid:

- C is at (-3, 0) (x=-3, y=0)

- A is at (-2, 4) (x=-2, y=4)

- G is at (-2, 0) (x=-2, y=0)

- E is at (-1, 2) (x=-1, y=2)

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

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

Now, for dilation with center origin, scale factor 2, the vector from origin to C is (-3,0), so image should be (-6,0), but G is (-2,0). Not matching. But the reason: corresponding sides (AC and EG) have different slopes (not parallel), so they are not dilations. So the correct option is "the corresponding sides are not parallel".

Wait, but the fourth option is "AC and EG are corresponding sides"—that's a statement, not a reason. So the reason is that corresponding sides (AC and EG) are not parallel (different slopes), so the polygons are not dilations.

Answer:

the corresponding sides are not parallel