In quadrilateral pqrs, the coordinates are p(0, 0), q(a + c, 0), r(2a + c, b), and s(a, b). How can you use coordinate geometry to show that the diagonals are perpendicular?.

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23-12-2022
Mathematics

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In quadrilateral pqrs, the coordinates are p(0, 0), q(a + c, 0), r(2a + c, b), and s(a, b). How can you use coordinate geometry to show that the diagonals are perpendicular?.

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mhanifa mhanifa · Answer 1 · 2022-12-25T13:01:46-05:00

Find the slopes and their product should be - 1 for the diagonals to be perpendicular.

The points p and q have zero y-coordinate, so they are on the y-axis.

The points r and s have same y-coordinate b, so they are on the parallel line to the y-axis.

The points p and r are opposite to each-other, similarly points s and q are opposite points. So the diagonals are pr and qs. This is also known from the order of vertices in the name of the quadrilateral.

Slopes are:

m(pr) = (b - 0)/(2a + c - 0) = b / (2a + c)
m(qs) = (b - 0) / (a - a - c) = - b/c

The product of slopes:

b / (2a + c) × ( - b / c) = - b² / (2ac + c²)

The diagonals are perpendicular if:

b² = 2ac + c²