Triangle XYZ is drawn on a coordinate plane with vertices X(0,4) Y(2,0) Z(4,7). if u dilate the triangle to form triangle X'Y'Z' using a magnitude of 1/2 and a center of dilation at (4,7), what are the coordinate points of the new vertices

To find the new coordinates of the dilated triangle \( X'Y'Z' \) with respect to the center of dilation at point \( (4,7) \) and a scale factor (magnitude) of \( \frac{1}{2} \), we will apply the dilation formula. The general formula for dilation of a point \( (x, y) \) about a center of dilation \( (c_x, c_y) \) with scale factor \( k \) is given by:

\[ (x', y') = (c_x + k(x - c_x), c_y + k(y - c_y)) \]

Step 1: Dilation of point \( X(0, 4) \)

• \( c_x = 4, c_y = 7, k = \frac{1}{2} \)
• For point \( X(0, 4) \):
  • \( x' = 4 + \frac{1}{2}(0 - 4) = 4 - 2 = 2 \)
  • \( y' = 7 + \frac{1}{2}(4 - 7) = 7 - \frac{3}{2} = 7 - 1.5 = 5.5 \)

So, point \( X' = (2, 5.5) \).

Step 2: Dilation of point \( Y(2, 0) \)

• For point \( Y(2, 0) \):
  • \( x' = 4 + \frac{1}{2}(2 - 4) = 4 + \frac{1}{2}(-2) = 4 - 1 = 3 \)
  • \( y' = 7 + \frac{1}{2}(0 - 7) = 7 - \frac{7}{2} = 7 - 3.5 = 3.5 \)

So, point \( Y' = (3, 3.5) \).

Step 3: Dilation of point \( Z(4, 7) \)

• For point \( Z(4, 7) \):
  • \( x' = 4 + \frac{1}{2}(4 - 4) = 4 + 0 = 4 \)
  • \( y' = 7 + \frac{1}{2}(7 - 7) = 7 + 0 = 7 \)

So, point \( Z' = (4, 7) \).

Summary of the new vertices:

• \( X'(2, 5.5) \)
• \( Y'(3, 3.5) \)
• \( Z'(4, 7) \)

Thus, the coordinates of the new vertices after dilation are \( X'(2, 5.5) \), \( Y'(3, 3.5) \), and \( Z'(4, 7) \).

suppose parallelogram ABCD is dilated using a magnitude of 2 and a center of dilation at -2,-1. what would be the corresponding ordered pair at point a

A(-3,1)

To dilate a point \(A(x, y)\) using a center of dilation \((c_x, c_y)\) and a scale factor \(k\), we use the following formula:

\[ (x', y') = (c_x + k(x - c_x), c_y + k(y - c_y)) \]

Given:

• Point \(A(-3, 1)\)
• Center of dilation \((-2, -1)\)
• Scale factor \(k = 2\)

Step 1: Calculate the new coordinates for point A

1. Substitute the values into the formula.

   • \(c_x = -2\)
   • \(c_y = -1\)
   • \(k = 2\)
   • \(x = -3\)
   • \(y = 1\)

2. Calculate the new x-coordinate \(x'\):

\[ x' = -2 + 2(-3 - (-2)) = -2 + 2(-3 + 2) = -2 + 2(-1) = -2 - 2 = -4 \]

3. Calculate the new y-coordinate \(y'\):

\[ y' = -1 + 2(1 - (-1)) = -1 + 2(1 + 1) = -1 + 2 \cdot 2 = -1 + 4 = 3 \]

Step 2: Write down the new coordinates of point A

The new coordinates of point \(A\) after dilation are:

\[ A' = (-4, 3) \]

Thus, the corresponding ordered pair at point \(A\) after dilation is \((-4, 3)\).