Question

select the correct answer.
the points a(-3, 4), b(3, 2), c(1, -4), and d(-5, -2) form quadrilateral abcd in the coordinate plane. what condition verifies that the diagonals are perpendicular?
a. the product of the slopes of the diagonals is -1.
b. the diagonals have different lengths.
c. the diagonals have the same length.
d. the product of the slopes of the diagonals is 1.
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Explanation:

Step1: Find slopes of diagonals

Diagonals are \(AC\) and \(BD\). For \(A(-3,4)\) and \(C(1,-4)\), slope \(m_{AC}=\frac{-4 - 4}{1 - (-3)}=\frac{-8}{4}=-2\). For \(B(3,2)\) and \(D(-5,-2)\), slope \(m_{BD}=\frac{-2 - 2}{-5 - 3}=\frac{-4}{-8}=\frac{1}{2}\).

Step2: Check product of slopes

Product of slopes: \(m_{AC}\times m_{BD}=(-2)\times\frac{1}{2}=-1\). Perpendicular lines have slope product \(-1\).

Answer:

A. The product of the slopes of the diagonals is -1