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 diagonals have the same length

b. the product of the slopes of the diagonals is -1

c. the diagonals have different lengths

d. the product of the slopes of the diagonals is 1

Explanation:

To determine if two lines (the diagonals here) are perpendicular, we use the property of slopes. If two non - vertical lines have slopes \(m_1\) and \(m_2\), they are perpendicular if and only if the product of their slopes \(m_1\times m_2=- 1\).

• Option A: The length of the diagonals being the same is a property related to whether the diagonals are congruent, not perpendicular. So A is incorrect.
• Option B: As per the slope property for perpendicular lines, if the product of the slopes of two lines is - 1, the lines are perpendicular. This is the correct condition for the diagonals to be perpendicular.
• Option C: The diagonals having different lengths has nothing to do with the perpendicularity of the diagonals. So C is incorrect.
• Option D: If the product of the slopes is 1, the lines are not perpendicular (they would be parallel only if the slopes are equal, but a product of 1 doesn't imply perpendicularity). So D is incorrect.

Answer:

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