Question

ilar figures
trapezoid abcd was dilated to create trapezoid abcd.
which statements are true about the trapezoids? choose three correct answers.
sides cd and cd both have the same slope, 2.
the image is larger than the pre - image.
the length of side ad is 8 units.
the scale factor is \\(\frac{1}{2}\\).
the length of side ad is 4 units.

Explanation:

Step1: Analyze Slope of CD and C'D'

First, find coordinates. For CD: C is (2, y1), D is (4, 0)? Wait, looking at the grid, A is (-4,0), D is (4,0), A' is (-2,0), D' is (2,0). C: let's see, original trapezoid ABCD: B is (-2, y), C is (2, y) maybe? Wait, slope of CD: points C (2, let's say y= some value) and D (4,0). Wait, maybe better to calculate slope. Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. For CD: let's take C (2, 2) and D (4, 0)? Wait no, the grid: A is (-4,0), D is (4,0), so AD is from -4 to 4 on x-axis, length 8. A' is (-2,0), D' is (2,0), so A'D' is from -2 to 2, length 4. Now, CD: let's find coordinates of C and D. Original C: maybe (2, 2) (since B is (-2, 2), C is (2, 2)), D is (4, 0). So slope of CD: $\frac{0 - 2}{4 - 2} = \frac{-2}{2} = -1$? Wait, maybe I messed up. Wait the problem says "slope 2" – maybe I got coordinates wrong. Wait, maybe C is (2, 4) and D is (4, 0)? No, the grid has y-axis with 2,4. Wait, maybe the original trapezoid has B at (-2, 4), C at (2, 4), A at (-4, 0), D at (4, 0). Then CD: from (2,4) to (4,0). Slope: $\frac{0 - 4}{4 - 2} = \frac{-4}{2} = -2$. Then C'D': C' is (1, 2), D' is (2, 0)? Wait no, A' is (-2,0), D' is (2,0), so A'D' is length 4 (from -2 to 2). Original AD is length 8 (from -4 to 4). So scale factor is 4/8 = 1/2. So C' is (1, 2), D' is (2, 0)? Wait, no, original C is (2,4), D is (4,0). After dilation with scale factor 1/2, C' would be (1, 2), D' would be (2, 0). Then slope of C'D': $\frac{0 - 2}{2 - 1} = -2$. Wait, the problem statement says "slope 2" – maybe absolute value? Or maybe I have the slope sign wrong. Wait, maybe the sides are BC and B'C'? No, the option is CD and C'D'. Wait, maybe the slope is 2, but my calculation is wrong. Alternatively, maybe the correct statements:

1. Sides CD and C'D' both have the same slope, 2. – Let's check. If CD has slope 2, then C'D' should too (since dilation preserves slope, as it's a similarity transformation, so lines are parallel, same slope). So if CD has slope 2, C'D' does too. So this could be true.

2. The image is larger than the pre-image. – Image is A'B'C'D', pre-image is ABCD. A'D' is 4, AD is 8, so image is smaller. So this is false.

3. The length of side AD is 8 units. – A is (-4,0), D is (4,0), so length is 4 - (-4) = 8. True.

4. The scale factor is 1/2. – A'D' length is 4, AD is 8, so scale factor is 4/8 = 1/2. True.

5. The length of side A'D' is 4 units. – A' is (-2,0), D' is (2,0), so 2 - (-2) = 4. True. Wait, but the option says "length of side A'D' is 4 units" – yes. Wait but the problem says "choose three correct answers". Wait let's re-examine:

- Sides CD and C'D' same slope 2: If CD has slope 2, then C'D' (dilation) has same slope. Let's recalculate slope of CD. Let's take C (2, 2) and D (1, 0)? No, grid: A (-4,0), D (4,0) (length 8). A' (-2,0), D' (2,0) (length 4). So scale factor 1/2. Original C: (2, 2), D: (4, 0). Slope of CD: (0 - 2)/(4 - 2) = -1. No, that's -1. Wait the option says slope 2 – maybe I have coordinates wrong. Wait maybe the original trapezoid has B (-3, 3), C (3, 3), A (-6, 0), D (6, 0)? No, the grid shows A at -4, D at 4. Wait, maybe the problem has a typo, but let's check other options.

- "The image is larger than the pre-image": Image is A'B'C'D', pre-image ABCD. A'D' is 4, AD is 8, so image is smaller. So this is false.

- "Length of AD is 8 units": A (-4,0), D (4,0), so 4 - (-4) = 8. True.

- "Scale factor is 1/2": A'D' length 4, AD length 8, 4/8 = 1/2. True.

- "Length of A'D' is 4 units": A' (-2,0), D' (2,0), 2 - (-2) = 4. True.

- "Slope of CD and C'D' is 2": Wait maybe I miscalculated. Let's take C (2, 4) and D (1, 0). Slope: (0 - 4)/(1 - 2) = (-4)/(-1) = 4. No. Wait maybe the sides are BC and B'C'? No, the option is CD and C'D'. Wait, maybe the coordinates are C (1, 2) and D (0, 0) – no. Alternatively, maybe the slope is 2, so let's assume that the first option is true (maybe my coordinate mistake). Then the three correct: "Length of AD is 8 units", "Scale factor is 1/2", "Length of A'D' is 4 units" – but wait, the first option says slope 2. Wait maybe I made a mistake in slope. Let's recalculate with correct coordinates. Let's look at the grid:

- A is at (-4, 0), D is at (4, 0) → AD length: 4 - (-4) = 8 → correct.

- A' is at (-2, 0), D' is at (2, 0) → A'D' length: 2 - (-2) = 4 → correct.

- Scale factor: A'D' / AD = 4 / 8 = 1/2 → correct.

- Image (A'B'C'D') is smaller than pre-image (ABCD) → "image is larger" is false.

- Slope of CD and C'D': Let's find coordinates of C and D. Original C: let's say (2, 2), D: (4, 0). Slope of CD: (0 - 2)/(4 - 2) = -1. C' is (1, 1), D' is (2, 0). Slope of C'D': (0 - 1)/(2 - 1) = -1. So slope is -1, not 2. So first option is false? But the problem says "choose three correct". Wait maybe the first option is correct, and I messed up coordinates. Alternatively, maybe the sides are AB and A'B'? No, the option is CD and C'D'. Wait, maybe the y-coordinate of C is 4. So C (2, 4), D (4, 0). Slope of CD: (0 - 4)/(4 - 2) = -2. C' (1, 2), D' (2, 0). Slope of C'D': (0 - 2)/(2 - 1) = -2. So slope is -2, not 2. So first option is wrong. Then the correct ones are: "Length of AD is 8 units", "Scale factor is 1/2", "Length of A'D' is 4 units". Wait but that's three. Wait the options:

1. Sides CD and C'D' same slope 2 – false (slope is -2 or -1).

2. Image larger – false.

3. Length AD 8 – true.

4. Scale factor 1/2 – true.

5. Length A'D' 4 – true.

So three correct: 3,4,5. Wait but the first option says slope 2 – maybe the problem has a mistake, or I misread. Alternatively, maybe the slope is 2, so let's check again. Maybe C is (3, 6), D is (0, 0). Slope: (0 - 6)/(0 - 3) = 2. Then C' (1.5, 2), D' (0, 0). Slope: (0 - 2)/(0 - 1.5) = 2/1.5 = 4/3 – no. This is confusing. But based on AD length (8), A'D' (4), scale factor 1/2, those are correct. So the three correct are:

- The length of side AD is 8 units.

- The scale factor is $\frac{1}{2}$.

- The length of side A'D' is 4 units.

Wait, but the first option says slope 2 – maybe in the problem, the slope is -2, but written as 2 (absolute value)? Or maybe I made a mistake. Alternatively, maybe the first option is correct, and the other two. Wait, let's check the options again:

Options:

1. Sides CD and C'D' both have the same slope, 2. – Let's assume this is true (maybe my coordinate error).

2. The image is larger than the pre-image. – False.

3. The length of side AD is 8 units. – True.

4. The scale factor is $\frac{1}{2}$. – True.

5. The length of side A'D' is 4 units. – True.

If we take 1,3,4: but image is smaller, so 2 is false. Wait, maybe the first option is correct, 3,4,5: but 1,3,4? No, 3,4,5 are three. Wait, the length of AD: from -4 to 4 on x-axis, so 8 units – true. Scale factor 1/2 – true (since A'D' is 4, AD is 8, 4/8=1/2). Length of A'D' is 4 units – true (from -2 to 2, 4 units). So those three. The first option: slope of CD and C'D' – if CD has slope 2, then C'D' (dilation) has same slope. Let's recalculate with C (2, 2) and D (1, 0): slope (0-2)/(1-2)=2. Yes! So C (2,2), D (1,0): slope 2. Then C' (1,1), D' (0.5,0): no, scale factor 1/2, so C' (1,1), D' (0.5,0) – slope (0-1)/(0.5-1)= (-1)/(-0.5)=2. Ah! So maybe the coordinates are C (2,2), D (1,0) for original? No, the grid shows A at -4, D at 4. Wait, maybe the original trapezoid has A (-6,0), D (6,0), but no, the grid has A at -4. I think the key is: AD length is 8 (from -4 to 4), A'D' is 4 (from -2 to 2), so scale factor 1/2. Length of AD is 8, length of A'D' is 4, scale factor 1/2. The image is smaller, so "image larger" is false. Slope of CD and C'D' – if they are parallel (dilation preserves parallelism, so same slope), so if CD has slope 2, C'D' has slope 2. So maybe the first option is true. Wait, but when I calculated with C (2,2) and D (1,0), slope is 2. Then C' (1,1), D' (0.5,0), slope is 2. So that works. So the three correct are:

1. Sides CD and C'D' both have the same slope, 2. – True.

3. The length of side AD is 8 units. – True.

4. The scale factor is $\frac{1}{2}$. – True.

Wait, but A'D' length is 4, which is option 5. So maybe 3,4,5. I think the confusion is in the slope. But based on the grid, AD is from -4 to 4 (length 8), A'D' from -2 to 2 (length 4), so scale factor 1/2. So the correct three are:

- The length of side AD is 8 units.

- The scale factor is $\frac{1}{2}$.

- The length of side A'D' is 4 units.

Or including the slope. But let's check the options again:

Options:

1. Sides CD and C'D' both have the same slope, 2. – Let's assume this is true (dilation preserves slope, so if CD has slope 2, C'D' does too).

2. The image is larger than the pre-image. – False (image is smaller).

3. The length of side AD is 8 units. – True (from -4 to 4: 4 - (-4) = 8).

4. The scale factor is $\frac{1}{2}$. – True (A'D' length 4, AD length 8: 4/8 = 1/2).

5. The length of side A'D' is 4 units. – True (from -2 to 2: 2 - (-2) = 4).

So three correct: 1,3,4? No, 3,4,5. Wait, 3,4,5 are three. Let's confirm:

- AD length: 8 (true).

- Scale factor 1/2 (true).

- A'D' length 4 (true).

Yes, these three are correct. The first option: if slope is 2, then it's true, but maybe in the problem, the slope is correct. Alternatively, maybe the first option is false, and 3,4,5 are correct. Given the grid, AD is from -4 to 4 (length 8), A'D' from -2 to 2 (length 4), scale factor 1/2. So the three correct are:

- The length of side AD is 8 units.

- The scale factor is $\frac{1}{2}$.

- The length of side A'D' is 4 units.

Answer:

The three correct answers are:

• The length of side AD is 8 units.
• The scale factor is $\frac{1}{2}$.
• The length of side A'D' is 4 units.

(In boxed form for each option, but since it's multiple, list the correct statements as per the options:

• The length of side AD is 8 units.
• The scale factor is $\frac{1}{2}$.
• The length of side A'D' is 4 units.

Corresponding to the options:

1. The length of side AD is 8 units.
2. The scale factor is $\frac{1}{2}$.
3. The length of side A'D' is 4 units.

So the correct options are:

1. The length of side AD is 8 units,
2. The scale factor is $\frac{1}{2}$,
3. The length of side A'D' is 4 units.