Question

Use the image to answer the question y 10- 9- 8- 7- 6- 5 X < 4- 3 2- 1 10+ -10-9-8-7-6-5-4-3-2-10 XY is dilated with a scale factor of 2 and a center of dilation at the origin. Find points X' and Y' to prove that a dilation of 2 has a length that is twice the length of XY (1 point) X' is at (-6, 8) and Y' is at (-14, 4) X' is at (-1.5, 2) and Y' is at (-3.5, 1) X' is at (6, -8) and Y' is at (14, -4) X' is at (-4,1) and Y' is at (-12,5)

Answer 1

To find the points \(X'\) and \(Y'\) after a dilation with a scale factor of 2 and a center of dilation at the origin, we can apply the following formula for dilation:

\[ X' = (kx, ky) \]

where \((x, y)\) are the original coordinates of point \(X\) or \(Y\), and \(k\) is the scale factor.

Assuming the original points \(X\) and \(Y\) are given by their coordinates, we can establish the new coordinates after dilation. However, it's important to note that the original points weren't provided in your question.

If we had the coordinates of \(X\) and \(Y\), we could then find \(X'\) and \(Y'\). Since it looks like you're given options for \(X'\) and \(Y'\), we want to confirm which of those represents a dilation by a scale factor of 2.

Let’s analyze the options you provided:

1. \(X' = (-6, 8)\) and \(Y' = (-14, 4)\)
2. \(X' = (-1.5, 2)\) and \(Y' = (-3.5, 1)\)
3. \(X' = (6, -8)\) and \(Y' = (14, -4)\)
4. \(X' = (-4,1)\) and \(Y' = (-12,5)\)

To determine which pair correctly represents a dilation from the origin, we need the original coordinates. Without knowing the initial coordinates \(X=(x_1,y_1)\) and \(Y=(x_2,y_2)\), we cannot definitively calculate \(X'\) and \(Y'\).

However, assuming we are starting from points \(X\) and \(Y\) and calculating:

Let's check one for scale factor 2:

• If \(X = (x_1, y_1)\), then \(X' = (2x_1, 2y_1)\).
• If \(Y = (x_2, y_2)\), then \(Y' = (2x_2, 2y_2)\).

For Option 3, if we suppose \(X\) is at \( (3, -4) \) (as an example):

• \(X' = (6, -8)\) which is \( X = (3, -4) \) scaled by 2 gives \( (23, 2-4) = (6, -8) \). If we do a similar hypothetical calculation for one of the others, we have to make the task complete.

With more specific original points, we could compute which of the options is indeed correct by checking if they satisfy the dilation property.

To conclude, without explicit original coordinates for \(X\) and \(Y\), it is tough to identify the correct pair of points among the given options solely based on the dilation definition.

If one of the options represents a doubling of the lengths or coordinates of the original points, that would be the result of the dilation. Thus, please provide coordinates for X and Y points before dilation for a more accurate confirmation!