Question

suppose you dilate the rectangle A(1,1)B(0,2)C(2,4)D(3,3) on the coordinate plane using a magnitude of 1/2 and the center of dilation at (1,1). what would be the corresponding ordered pair to the current point D.

Answer 1

To perform a dilation of a point using a center of dilation, we can use the following formula for the coordinates of the dilated point:

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

where:
- \((x', y')\) are the coordinates of the dilated point,
- \((x, y)\) are the coordinates of the original point,
- \((x_c, y_c)\) are the coordinates of the center of dilation,
- \(k\) is the magnitude of the dilation.

In this case, your coordinates for point \(D\) are \(D(3, 3)\), the center of dilation is \((1, 1)\), and the magnitude of dilation is \(k = \frac{1}{2}\).

Now we can substitute the values into the dilation formula:
- \(x_c = 1\), \(y_c = 1\)
- \(x = 3\), \(y = 3\)
- \(k = \frac{1}{2}\)

Calculating the new coordinates:
\[
x' = 1 + \frac{1}{2}(3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2
\]

\[
y' = 1 + \frac{1}{2}(3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2
\]

Thus, the ordered pair for the point \(D\) after the dilation is \(D'(2, 2)\).