Question

The points D(-2,2), E(2,5), and F(5,1) form ADEF in the coordinate plane.

What condition verifies that the triangle is a right triangle?
OA. Three of the sides of the triangle are equal.
B. The product of the slopes of two of the sides is −1.
OC. The product of the slopes of two of the sides is 1.
OD. Two of the sides of the triangle are equal.

Answer 1

The product of the slopes of two of the sides is -1.

Explanation:

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Answer 2

B. The product of the slopes of two of the sides is −1.

Explanation:

You want to know what verifies that points D(-2,2), E(2,5), and F(5,1) are the vertices of a right triangle.

Slope

The slopes of the line segments of interest are ...

m = (y2 -y1)/(x2 -x1)

m1 = (5 -2)/(2 -(-2)) = 3/4 . . . . . . slope of DE

m2 = (1 -5)/(5 -2) = -4/3 . . . . . . . . slope of EF

Right angle

The product of the slopes of these lines is ...

m1·m2 = (3/4)(-4/3) = -12/12) = -1

This product means the lines are perpendicular. A triangle with perpendicular sides is a right triangle.

We know ∆DEF is a right triangle because the product of the slopes of two of the sides is -1.

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