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 product of the slopes of the diagonals is -1.

d. the diagonals have the same length.

Explanation:

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

• Option A: If the product of slopes is 1, the lines are parallel (for non - vertical lines with slopes \(m_1 = m_2\)) or have a special non - perpendicular relationship, so A is incorrect.
• Option B: The length of the diagonals has nothing to do with their perpendicularity. Perpendicularity is a property related to the angle between the lines (determined by slopes), not their lengths, so B is incorrect.
• Option C: By the definition of perpendicular lines in the coordinate plane, 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 D: Having the same length is a property of congruent segments, not a condition for perpendicularity, so D is incorrect.

Answer:

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