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
the slope of eg is
the polygons are not dilations of each other because

Explanation:

Step1: Find coordinates of points for AC

First, identify coordinates of A and C. From the graph, A is at (-2, 4), C is at (-3, 0). Wait, no, let's check again. Wait, looking at the grid: C is at (-3, 0)? Wait, no, the x-axis: C is at (-3, 0)? Wait, the grid lines: let's see, A is at (-2, 4)? Wait, no, maybe A is at (-2, 4)? Wait, no, the top trapezoid: A is at (-2, 4)? Wait, no, let's look at the x-coordinates. Let's list the points:

Point A: Let's see, the top left vertex. From the grid, x=-2, y=4? Wait, no, maybe A is (-2, 4), B is (2, 4), C is (-3, 0), D is (3, 0)? Wait, no, the bottom left: G is at (-2, 0)? Wait, no, the x-axis: C is at (-3, 0), G is at (-2, 0), E is at (-1, 2), F is at (1, 2), H is at (2, 0), D is at (3, 0). Wait, maybe A is (-2, 4), B is (2, 4), C is (-3, 0), D is (3, 0), E is (-1, 2), F is (1, 2), G is (-2, 0), H is (2, 0). Wait, no, the original trapezoid is ABDC, so A, B, D, C. So A(-2, 4), B(2, 4), D(3, 0), C(-3, 0)? Wait, no, the grid: x from -4 to 4, y from -4 to 4. Let's check the coordinates properly.

Point A: Let's count the grid squares. From the origin (0,0), moving left 2 units (x=-2) and up 4 units (y=4), so A(-2, 4). Point B: right 2 units (x=2), up 4 units (y=4), so B(2, 4). Point C: left 3 units (x=-3), y=0, so C(-3, 0). Point D: right 3 units (x=3), y=0, so D(3, 0). Then the smaller trapezoid: E(-1, 2), F(1, 2), G(-2, 0), H(2, 0). Wait, no, G is at (-2, 0), H at (2, 0), E at (-1, 2), F at (1, 2).

Now, slope of AC: points A(-2, 4) and C(-3, 0). The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. So $m_{AC} = \frac{0 - 4}{-3 - (-2)} = \frac{-4}{-1} = 4$. Wait, no, wait: A is (-2, 4), C is (-3, 0)? Wait, no, maybe I got the coordinates wrong. Wait, looking at the graph, the left side of the big trapezoid: from C (on x-axis, x=-3) up to A. Wait, maybe C is at (-3, 0), A is at (-2, 4). So the change in y: 4 - 0 = 4, change in x: -2 - (-3) = 1. So slope is 4/1 = 4? Wait, no, that seems steep. Wait, maybe A is (-2, 4), C is (-3, 0)? Wait, no, maybe C is at (-3, 0), A is at (-2, 4). So the slope is (4 - 0)/(-2 - (-3)) = 4/1 = 4. Wait, but maybe I mixed up the points. Wait, the big trapezoid: A and B are on y=4, C and D on y=0. So A(-2, 4), B(2, 4), C(-3, 0), D(3, 0). Then the smaller trapezoid: E(-1, 2), F(1, 2), G(-2, 0), H(2, 0). So EG is from E(-1, 2) to G(-2, 0). Slope of EG: (0 - 2)/(-2 - (-1)) = (-2)/(-1) = 2. Wait, but let's check again.

Wait, maybe the coordinates are: A(-2, 4), C(-3, 0)? No, maybe C is at (-3, 0), A is at (-2, 4). So the slope of AC: (4 - 0)/(-2 - (-3)) = 4/1 = 4. Then slope of EG: E(-1, 2) to G(-2, 0). (0 - 2)/(-2 - (-1)) = (-2)/(-1) = 2. Wait, but the problem says "the polygons are not dilations of each other because..." Dilation requires that the lines from the center (origin) pass through corresponding points, and the scale factor is consistent. Let's check the distances from the origin. For point A(-2, 4), the distance from origin is $\sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$. The corresponding point E(-1, 2): distance is $\sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$. So the scale factor from A to E is $\sqrt{5}/(2\sqrt{5}) = 1/2$, which is correct (scale factor 2, so dilation from E to A would be 2). But wait, the slope of AC and EG: if dilation with scale factor 2, the slopes should be the same, because dilation preserves slope (since it's a linear transformation). Wait, but earlier calculation: slope of AC is 4, slope of EG is 2? That can't be. So maybe I got the coordinates wrong.

Wait, let's re-examine the graph. Let's look at the big trapezoid: A is at (-2, 4), B at (2, 4), C at (-3, 0), D at (3, 0). The small trapezoid: E at (-1, 2), F at (1, 2), G at (-2, 0), H at (2, 0). Wait, no, G is at (-2, 0), which is on the x-axis, same as C and D? Wait, no, C is at (-3, 0), G at (-2, 0), H at (2, 0), D at (3, 0). So the big trapezoid has bases AB (length 4: from -2 to 2 on x, y=4) and CD (length 6: from -3 to 3 on x, y=0). The small trapezoid has bases EF (length 2: from -1 to 1 on x, y=2) and GH (length 4: from -2 to 2 on x, y=0). Now, dilation with scale factor 2 from origin: the image of E(-1,2) should be A(-2,4) (since -1*2=-2, 2*2=4), which is correct. The image of G(-2,0) should be C(-4,0), but C is at (-3,0), not (-4,0). Ah! So that's the problem. So the center of dilation is origin, scale factor 2. So the image of G(-2,0) should be (-2*2, 0*2)=(-4,0), but C is at (-3,0), not (-4,0). Therefore, the polygons are not dilations because the corresponding points do not lie on the same line from the center (origin), or the scale factor is not consistent. Wait, let's check the slope of AC and EG again.

Wait, maybe A is (-2, 4), C is (-3, 0)? No, if G is (-2,0), then the image of G under dilation (scale 2, origin) should be (-4,0), but C is at (-3,0). So the line from origin to G is y=0 (x-axis), and line from origin to C is also y=0, but G is at (-2,0), C at (-3,0): the distance from origin to G is 2, to C is 3, so scale factor would be 3/2, not 2. So the scale factor is not consistent. Also, the slope of AC: let's recalculate. A(-2,4), C(-3,0). Slope: (0-4)/(-3 - (-2)) = (-4)/(-1) = 4. Slope of EG: E(-1,2), G(-2,0). Slope: (0-2)/(-2 - (-1)) = (-2)/(-1) = 2. Wait, but if dilation, the slopes should be equal because dilation is a linear transformation (preserves slope). But here, slopes are 4 and 2, which are different, so that's another reason. Wait, but maybe my coordinate assignment is wrong.

Wait, maybe A is (-2, 4), C is (-3, 0)? No, let's look at the graph again. The big trapezoid: left side from C (x=-3, y=0) up to A (x=-2, y=4). The small trapezoid: left side from G (x=-2, y=0) up to E (x=-1, y=2). So slope of AC: (4-0)/(-2 - (-3)) = 4/1 = 4. Slope of EG: (2-0)/(-1 - (-2)) = 2/1 = 2. So slopes are 4 and 2, not equal. Also, the lines from origin: line OA (O is origin) goes to A(-2,4), line OE goes to E(-1,2) (which is correct, since E is the image of A under scale factor 1/2). Line OG goes to G(-2,0), line OC should go to C(-3,0), but OG is (-2,0), OC is (-3,0): the ratio of distances is 3/2, not 2. So the corresponding points (G and C) do not lie on the same line from the center (origin) with scale factor 2. Therefore, the polygons are not dilations because the corresponding points do not lie on the same line from the center (origin), or the scale factor is not consistent.

Now, let's correct the slope calculations. Wait, maybe I mixed up the points. Let's take A(-2, 4) and C(-3, 0): slope is (4 - 0)/(-2 - (-3)) = 4/1 = 4. EG: E(-1, 2) and G(-2, 0): slope is (2 - 0)/(-1 - (-2)) = 2/1 = 2. Wait, but that's not matching. Wait, maybe A is (-2, 4), C is (-3, 0) is wrong. Maybe C is (-3, 0), A is (-2, 4). Then the slope is 4. EG: E(-1, 2), G(-2, 0): slope is 2. Then the third statement: the polygons are not dilations because the corresponding sides are not parallel (slopes are different) or the corresponding points do not lie on the same line from the center.

Wait, the key here is that for dilation, the lines connecting corresponding points must pass through the center (origin), and the scale factor must be constant. Let's check the coordinates:

- E(-1,2) and A(-2,4): A is E scaled by 2 (correct, -1*2=-2, 2*2=4).
- F(1,2) and B(2,4): B is F scaled by 2 (correct, 1*2=2, 2*2=4).
- G(-2,0) and C(-3,0): C is not G scaled by 2 (G*2=-4,0, but C is -3,0).
- H(2,0) and D(3,0): D is not H scaled by 2 (H*2=4,0, but D is 3,0).

Therefore, the corresponding points G and C, H and D do not lie on the same line from the origin (they are on the x-axis, but the distances from origin are 2 and 3 for G and C, 2 and 3 for H and D, so scale factor 3/2, not 2). Hence, the polygons are not dilations because the corresponding vertices do not lie on the same line from the center (origin), or the scale factor is not consistent.

Now, slope of AC: points A(-2,4) and C(-3,0). Slope = (0-4)/(-3 - (-2)) = (-4)/(-1) = 4.

Slope of EG: points E(-1,2) and G(-2,0). Slope = (0-2)/(-2 - (-1)) = (-2)/(-1) = 2.

Wait, but maybe I made a mistake in coordinates. Let's check again. Maybe C is at (-3, 0), A is at (-2, 4). So AC: from (-3,0) to (-2,4). So slope is (4-0)/(-2 - (-3)) = 4/1 = 4. EG: from E(-1,2) to G(-2,0). Slope is (0-2)/(-2 - (-1)) = (-2)/(-1) = 2.

Then, the reason the polygons are not dilations: the corresponding points (like G and C) do not lie on the same line from the center (origin) with the correct scale factor. Or, the slopes of corresponding sides are not equal (since dilation preserves slope, but here AC has slope 4 and EG has slope 2, which are different).

So summarizing:

1. Slope of AC: 4.

2. Slope of EG: 2.

3. The polygons are not dilations because the corresponding vertices do not lie on the same line from the center (origin), or the scale factor is not consistent (e.g., the image of G under dilation with scale factor 2 from origin is (-4,0), but C is at (-3,0), so they are not on the same line from the origin with scale factor 2).

Answer:

The slope of AC is $\boldsymbol{4}$.
The slope of EG is $\boldsymbol{2}$.
The polygons are not dilations of each other because the corresponding vertices do not lie on the same line from the center (origin) (or the scale factor is not consistent, or the slopes of corresponding sides are not equal).