To determine the type of quadrilateral ABCD with vertices [tex]\(A(10, 2)\)[/tex], [tex]\(B(2, -4)\)[/tex], [tex]\(C(-4, 4)\)[/tex], and [tex]\(D(4, 10)\)[/tex], let's follow these steps:
1. Calculate the lengths of all sides:
- Using the distance formula:
[tex]\[
AB = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2}
\][/tex]
[tex]\[
BC = \sqrt{(C_x - B_x)^2 + (C_y - B_y)^2}
\][/tex]
[tex]\[
CD = \sqrt{(D_x - C_x)^2 + (D_y - C_y)^2}
\][/tex]
[tex]\[
DA = \sqrt{(A_x - D_x)^2 + (A_y - D_y)^2}
\][/tex]
2. Calculate the slopes of all sides:
- Using the slope formula:
[tex]\[
m_{AB} = \frac{B_y - A_y}{B_x - A_x}
\][/tex]
[tex]\[
m_{BC} = \frac{C_y - B_y}{C_x - B_x}
\][/tex]
[tex]\[
m_{CD} = \frac{D_y - C_y}{D_x - C_x}
\][/tex]
[tex]\[
m_{DA} = \frac{A_y - D_y}{A_x - D_x}
\][/tex]
3. Check the properties based on calculated distances and slopes:
- Square: All four sides are equal, and adjacent sides are perpendicular.
- Rectangle: Opposite sides are equal, and adjacent sides are perpendicular.
- Parallelogram: Opposite sides are equal and parallel, but not necessarily perpendicular.
- Trapezoid: Only one pair of opposite sides are parallel.
Upon verifying all these conditions with the calculated values:
- The lengths of all sides (results not shown here) indicate the four sides are equal.
- The slopes of adjacent sides are perpendicular (product of slopes is -1).
Therefore, quadrilateral ABCD satisfies the conditions for a square with all sides equal and adjacent sides perpendicular.
Conclusion:
The correct statement is:
- ABCD is a square with perpendicular adjacent sides.
Thus, the correct answer is:
[tex]\[ \boxed{ABCD \text{ is a square with perpendicular adjacent sides.}} \][/tex]
