To determine which condition verifies that the diagonals of quadrilateral [tex]\(ABCD\)[/tex] are perpendicular, we need to examine the slopes of the diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex]. Here, we will find the respective slopes first and then analyze the product of those slopes.
1. Identifying the Points:
- [tex]\(A(-3, 4)\)[/tex]
- [tex]\(C(1, -4)\)[/tex]
- [tex]\(B(3, 2)\)[/tex]
- [tex]\(D(-5, -2)\)[/tex]
2. Calculating the Slope of Diagonal [tex]\(AC\)[/tex]:
The slope [tex]\(m_{AC}\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For points [tex]\(A(-3, 4)\)[/tex] and [tex]\(C(1, -4)\)[/tex]:
[tex]\[
m_{AC} = \frac{-4 - 4}{1 + 3} = \frac{-8}{4} = -2
\][/tex]
3. Calculating the Slope of Diagonal [tex]\(BD\)[/tex]:
For points [tex]\(B(3, 2)\)[/tex] and [tex]\(D(-5, -2)\)[/tex]:
[tex]\[
m_{BD} = \frac{-2 - 2}{-5 - 3} = \frac{-4}{-8} = \frac{1}{2} = 0.5
\][/tex]
4. Calculating the Product of Slopes:
To verify if the diagonals are perpendicular, we check the product of the slopes:
[tex]\[
m_{AC} \times m_{BD} = -2 \times 0.5 = -1
\][/tex]
5. Conclusion:
The product of the slopes of the diagonals is [tex]\(-1\)[/tex]. When the product of the slopes of two lines is [tex]\(-1\)[/tex], it means that the lines are perpendicular. Therefore, the condition that verifies the diagonals of quadrilateral [tex]\(ABCD\)[/tex] are perpendicular is that the product of the slopes of the diagonals is [tex]\(-1\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\text{D. The product of the slopes of the diagonals is -1.}}
\][/tex]
