Question

Dilate the triangle using a magnitude of 1/2 and a center of dilation at (2,2) to form triangle A'B'C'. What are the coordinates of the dilated triangle? (A is 5,5) (B is 1,7) (C is 3,2)

Answer 1

To dilate a triangle with a center of dilation and a scale factor, you can use the following formula for each vertex:

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

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

In this case:
- Center of dilation, \(C(2, 2)\)
- Scale factor, \(k = \frac{1}{2}\)

We will apply the dilation to each vertex of the triangle.

1. **Dilating point A(5, 5)**:
\[
x' = 2 + \frac{1}{2}(5 - 2) = 2 + \frac{1}{2} \times 3 = 2 + 1.5 = 3.5
\]
\[
y' = 2 + \frac{1}{2}(5 - 2) = 2 + \frac{1}{2} \times 3 = 2 + 1.5 = 3.5
\]
So, the new coordinates for point A' are (3.5, 3.5).

2. **Dilating point B(1, 7)**:
\[
x' = 2 + \frac{1}{2}(1 - 2) = 2 + \frac{1}{2} \times (-1) = 2 - 0.5 = 1.5
\]
\[
y' = 2 + \frac{1}{2}(7 - 2) = 2 + \frac{1}{2} \times 5 = 2 + 2.5 = 4.5
\]
So, the new coordinates for point B' are (1.5, 4.5).

3. **Dilating point C(3, 2)**:
\[
x' = 2 + \frac{1}{2}(3 - 2) = 2 + \frac{1}{2} \times 1 = 2 + 0.5 = 2.5
\]
\[
y' = 2 + \frac{1}{2}(2 - 2) = 2 + \frac{1}{2} \times 0 = 2 + 0 = 2
\]
So, the new coordinates for point C' are (2.5, 2).

In conclusion, after dilating triangle ABC by a factor of \( \frac{1}{2} \) with a center at (2, 2), the coordinates of the dilated triangle A'B'C' are:
- A'(3.5, 3.5)
- B'(1.5, 4.5)
- C'(2.5, 2)