Answer :
Answer:
To prove that the quadrilateral with vertices \( A(-2, 2) \), \( B(2, -2) \), \( C(4, 2) \), and \( D(2, 4) \) is a trapezoid with perpendicular diagonals, follow these steps:
1. **Calculate the slopes of the sides:**
The slope of a line segment \((x_1, y_1)\) to \((x_2, y_2)\) is given by:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
- **Slope of \(AB\):**
\[
\text{slope of } AB = \frac{-2 - 2}{2 - (-2)} = \frac{-4}{4} = -1
\]
- **Slope of \(BC\):**
\[
\text{slope of } BC = \frac{2 - (-2)}{4 - 2} = \frac{4}{2} = 2
\]
- **Slope of \(CD\):**
\[
\text{slope of } CD = \frac{4 - 2}{2 - 4} = \frac{2}{-2} = -1
\]
- **Slope of \(DA\):**
\[
\text{slope of } DA = \frac{2 - 4}{-2 - 2} = \frac{-2}{-4} = \frac{1}{2}
\]
2. **Determine if the quadrilateral is a trapezoid:**
A trapezoid has exactly one pair of parallel sides. Check if any two sides are parallel by comparing their slopes.
- **Slope of \(AB\) and \(CD\):** Both are \(-1\). So, \(AB \parallel CD\).
- **Slope of \(BC\) and \(DA\):** \(2\) and \(\frac{1}{2}\) respectively. They are not parallel.
Since \(AB\) is parallel to \(CD\) and there is exactly one pair of parallel sides, the quadrilateral is a trapezoid.
3. **Calculate the slopes of the diagonals:**
Diagonals are \(AC\) and \(BD\).
- **Slope of \(AC\):**
\[
\text{slope of } AC = \frac{2 - 2}{4 - (-2)} = \frac{0}{6} = 0
\]
(horizontal line)
- **Slope of \(BD\):**
\[
\text{slope of } BD = \frac{4 - (-2)}{2 - 2} = \frac{6}{0}
\]
(undefined, vertical line)
Since one diagonal is horizontal and the other is vertical, they are perpendicular.
**Conclusion:**
The quadrilateral with vertices \( A(-2, 2) \), \( B(2, -2) \), \( C(4, 2) \), and \( D(2, 4) \) is a trapezoid with perpendicular diagonals.
Step-by-step explanation:
