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Question

A coordinate plane's axes range from negative 1 to 4 by 1-unit increments. 4 points forming a rectangle are plotted. They are labeled upper A, upper B, upper C, and upper D, respectively.

Suppose you dilate the rectangle on the coordinate plane using a magnitude of 12
and the center of dilation at (1,1)
. What would be the corresponding ordered pair to the current point D
?

(1 point)
The new ordered pair for point D′
would be at (
,
).
Remaining Attempts : 3
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Original coordinaes for D are (3,3)

Answer 1

To find the new coordinates for point D after dilation, we can use the formula for dilation:

\[ D' = \left( k(x - x_c) + x_c, k(y - y_c) + y_c \right) \]

where \((x_c, y_c)\) is the center of dilation, \((x, y)\) is the original point, and \(k\) is the scale factor (magnitude of dilation).

Given:

• Original coordinates of D: \((3, 3)\)
• Center of dilation: \((1, 1)\)
• Magnitude of dilation: \(k = 12\)

Now, we can plug these into the formula:

1. Calculate the new x-coordinate: \[ x' = 12(3 - 1) + 1 = 12 \cdot 2 + 1 = 24 + 1 = 25 \]

2. Calculate the new y-coordinate: \[ y' = 12(3 - 1) + 1 = 12 \cdot 2 + 1 = 24 + 1 = 25 \]

Thus, the new coordinates of point D' after dilation will be: \[ D' = (25, 25) \]

The corresponding ordered pair for point D' is (25, 25).

Answer 2

thats incorrect

Answer 3

I apologize for the incorrect calculation. Let’s reassess the dilation step-by-step using the correct formula.

The dilation formula is:

\[ D' = \left( k(x - x_c) + x_c, k(y - y_c) + y_c \right) \]

Where:

• \( (x_c, y_c) \) is the center of dilation,
• \( (x, y) \) is the original point,
• \( k \) is the magnitude of dilation.

Given:

• Original coordinates of D \((x, y)\): \((3, 3)\)
• Center of dilation \((x_c, y_c)\): \((1, 1)\)
• Magnitude of dilation \(k = 12\).

Now, let’s substitute the values:

1. Calculate the new x-coordinate: \[ x' = 12(3 - 1) + 1 = 12 \cdot 2 + 1 = 24 + 1 = 25 \]

2. Calculate the new y-coordinate: \[ y' = 12(3 - 1) + 1 = 12 \cdot 2 + 1 = 24 + 1 = 25 \]

This results again in D' being at \( (25, 25) \), which is quite far from the coordinate plane you mentioned earlier.

Let's use another method to verify. The dilation of a point can also be visualized as moving it away from or toward the center of dilation.

Steps to find the new coordinates:

1. Find the distance from the center of dilation to point D:

   • The coordinates of D are \((3, 3)\) and the center of dilation is \((1, 1)\).
   • The vector from the center to D is: \[ (3-1, 3-1) = (2, 2) \]

2. Dilate this vector by the scale factor:

   • The original distance is \( (2, 2) \) which we scale by 12:
   • The new vector will be: \[ (12 \cdot 2, 12 \cdot 2) = (24, 24) \]

3. Add this new vector to the center of dilation:

   • Thus, the coordinates will be: \[ (1 + 24, 1 + 24) = (25, 25) \]

I realize this does not seem to fit within the original coordinates range you provided.

Resolution Steps:

Let’s further analyze how we should consider the center to work properly.

When using a dilation factor of 12 from the center \((1, 1)\):

• The distance from \((1, 1)\) to \((3, 3)\) is \(\sqrt{(3-1)^2 + (3-1)^2}\) which is not directly used here in coordinates.

To avoid this error, may I know if there’s more information about the scale factor or adjustments to be correctly brought out?

Final Take:

If we'd consider it as a standard scale from common factors, the direct re-evaluation yields that point D should remain succinct about its transformations without overwhelming the equations.

If indeed the distance considered was relative thereby, please clarify or review the prospective scope! Thank you. The correct resultant should yield something distinctly confined if otherwise!