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a -1,-1 b-1,1 c2,3 d2,-1 A -2,-2 B -2,2 C 4,6 D 4,-2

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

3
3

12
Start Fraction 1 over 2 End Fraction

1

3 answers

To determine the magnitude of dilation of polygon ABCD to form polygon A'B'C'D', we first need to find the distance of at least one point from the origin before and after the dilation.

First, let's calculate the distance of point A (-2, -2) from the origin (0, 0):

\[ \text{Distance of A} = \sqrt{(-2 - 0)^2 + (-2 - 0)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

Next, we calculate the distance of point A' (A prime) after dilation, which is point A (2, 2):

\[ \text{Distance of A'} = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

Now, to find the magnitude of the dilation, we need to compare the distances from the origin before and after:

If the distance from the origin to A was \( d_1 = 2\sqrt{2} \) and the distance from the origin to A' is \( d_2 = 2\sqrt{2} \), we observe that:

The dilation factor (magnitude of dilation) is given by:

\[ \text{Dilation Factor} = \frac{d_2}{d_1} \]

Since both distances are equal (\( d_2 = d_1 \)), the dilation factor is:

\[ \text{Dilation Factor} = \frac{2\sqrt{2}}{2\sqrt{2}} = 1 \]

So it appears that there was no dilation, or the context of the question might be incorrect since all distances calculated show unity. However, we can compare the coordinates of the points as:

  1. A (-2, -2) → A' (2, 2)
  2. B (-2, 2) → B' (2, -2)
  3. C (4, 6) → C' (2, 3)
  4. D (4, -2) → D' (2, -1)

Notably, each coordinate of A, B, C, and D has been transformed to similar coordinates of A', B', C', and D'.

If we assumed the dilation factor was based on a certain point (which it may not have been), I sense the error lies in defining the location of A' and others correctly, but the initial points were stretched, especially at coordinates possibly given by maximum x or y.

The correct answer, based on analyzing scaling with respect to the origin,

Based on simplification, magnitude of the dilation might be determined as

The dilation magnitude is 1, however if you provided the values above as an operation of coordinates, could indicate sprint towards approximation to the point set approaching values reaching the larger dilation potentially back to options given.

Simply stated never resulted in a clear stretch since all values drawn to the coordinates themselves would likely end to imply just regular operations available.

If confused regarding the prime's measures available or specified. Not any right out explanation of the coordinates provided yield towards varying from what could be reflective at distances or required.

so is 1 the answer

I apologize for any confusion in my previous response. Let's clarify the concept of dilation in your problem.

Dilation Explanation

In dilation with respect to the origin, the points of the polygon are multiplied by a dilation factor \( k \). If the original point is \((x, y)\), the diluted point will be \((kx, ky)\).

Given Points

Let's summarize the points for both polygons again:

Polygon ABCD:

  • A: (-2, -2)
  • B: (-2, 2)
  • C: (4, 6)
  • D: (4, -2)

Polygon A'B'C'D':

  • A': (2, 2)
  • B': (2, -2)
  • C': (2, 3)
  • D': (2, -1)

To find the dilation factor, we calculate the distance from the origin for a point in ABCD and its corresponding point in A'B'C'D'.

Example with Point A

  1. Distance of A (-2, -2) from the origin:

    \[ \text{Distance} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

  2. Distance of A' (2, 2) from the origin:

    \[ \text{Distance} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

Dilation Factor Calculation

The dilation factor \( k \) is calculated by comparing the distances:

\[ k = \frac{\text{Distance of A'}}{\text{Distance of A}} = \frac{2\sqrt{2}}{2\sqrt{2}} = 1 \]

Conclusion

Based on your points, the dilation factor is indeed 1, which means there is no change in size (the original polygon and the dilated polygon are the same size).

So the answer to the question about the magnitude of dilation is 1. If your options were different and did not include 1, please double-check the available choices or point coordinates.