Question

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

Answer 1

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)

Answer 2

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 \( Z'(4, 7) \).

Answer 3

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 \((-4, 3)\).