Use the image to answer the question.

Question

Smaller polygon upper A upper B upper C upper D is graphed within larger polygon upper A prime upper B prime upper C prime upper D prime on a coordinate plane.

In the image, polygon ABCD
is dilated to form polygon A′B′C′D′
with respect to the origin. What is the magnitude of dilation?

(1 point)
Responses

2
2

12
Start Fraction 1 over 2 End Fraction

1
1

3
3
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Answer 1

To determine the magnitude of dilation from polygon ABCD to polygon A′B′C′D′, we need to identify a corresponding pair of points from both polygons and analyze their coordinates.

The magnitude of dilation can be calculated using the formula:

\[ \text{Dilation Factor} = \frac{\text{Distance from origin to point A'}}{\text{Distance from origin to point A}} \]

Without the specific coordinates, I can't provide an exact numerical answer. However, you can apply this formula to any corresponding points from the two polygons to find the dilation factor.

Here is how to solve it step by step:

1. Identify the coordinates of a point in the original polygon (e.g., A).
2. Identify the coordinates of the corresponding point in the dilated polygon (e.g., A′).
3. Calculate the distance from the origin (0,0) to each point using the distance formula \( \sqrt{x^2 + y^2} \).
4. Divide the distance to the dilated point by the distance to the original point.

For example, if point A is at (x, y) and A′ is at (kx, ky), where k is the factor of dilation, you would find that:

\[ \text{Dilation Factor} = k \]

Using the choices you provided (2, 1/2, 1, 3), you can calculate and compare to find the correct option.

If you have the coordinates for points A and A′, please provide them, and I can help you with the calculations.

Answer 2

coordinates are C'(4,6) C is (2,3) B'(-2,2) B is (-1,1) A is (-1,-1) A" (-2,-2) nD is (2,-1) D'(4,-2)

Answer 3

To determine the dilation factor when polygon ABCD is dilated to polygon A'B'C'D', we can analyze the pairs of corresponding points you provided.

Let's calculate the distances to the origin for points A, A', B, B', C, C', D, and D':

1. For point A (-1, -1): \[ \text{Distance to origin} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

2. For point A' (-2, -2): \[ \text{Distance to origin} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

3. For point B (-1, 1): \[ \text{Distance to origin} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

4. For point B' (-2, 2): \[ \text{Distance to origin} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

5. For point C (2, 3): \[ \text{Distance to origin} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

6. For point C' (4, 6): \[ \text{Distance to origin} = \sqrt{(4)^2 + (6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]

7. For point D (2, -1): \[ \text{Distance to origin} = \sqrt{(2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

8. For point D' (4, -2): \[ \text{Distance to origin} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]

Dilation Factor Calculation:

Now, let's calculate the dilation factor using point A and A' as an example:

\[ \text{Dilation Factor} = \frac{\text{Distance to A'}}{\text{Distance to A}} = \frac{2\sqrt{2}}{\sqrt{2}} = 2 \]

You can check the same with points B and B', C and C', and D and D':

\[ \frac{2\sqrt{2}}{\sqrt{2}} = 2 \quad \text{(similar for B)} \] \[ \frac{2\sqrt{13}}{\sqrt{13}} = 2 \quad \text{(similar for C)} \] \[ \frac{2\sqrt{5}}{\sqrt{5}} = 2 \quad \text{(similar for D)} \]

Conclusion:

The magnitude of dilation from polygon ABCD to polygon A'B'C'D' is 2.