In a coordinate plane, two nonvertical lines are parallel if and only if their slopes are equal.
In a coordinate plane two nonvertical lines are perpendicular if and only if the product of their slopes is -1.
Consider the perpendicular lines D and E in the coordinate plane, that form a right triangle with the x-axis.
Using the points on each line, the slopes of D and E can be written as follows. mD=acandmE=b-c Multiplying the slopes gives their product. ac⋅b-c=ab-c2 Notice that the product equals -1 if and only if the following is true. ab-c2=-1⇔c2=ab⇔c=ab The Geometric Mean Altitude Theorem states that the altidtude of a right triangle is equal to the geometric mean of the segments the altitude creates with the hypotenuse. Notice that c is the altitude of the right triangle in the diagram. Also, c divides the hypotenuse into the segments a and b. Thus, c=ab is a true statement. Therefore, the product of the slopes of two perpendicular lines equals -1.
mD⋅mE=-1The points P(0.8,-0.6), Q(-1.6,4.2), R(1.2,5.6), and S(3.6,0.8) are vertices of a quadrilateral. Prove that quadrilateral is a rectangle.
To begin, we can graph the quadrilateral in a coordinate plane.
If quadrilateral PQRS is a rectangle, both pairs of opposite sides will be parallel and all pairs of adjacent sides will be perpendicular. Thus, we must show that
Line segment | Slope formula | Simplified | Slope |
---|---|---|---|
PQ | -1.6−0.84.2−(-0.6) | -2.44.8 | mPQ=-2 |
QR | 1.2−(-1.6)5.6−4.2 | 2.81.4 | mQR=0.5 |
RS | 3.6−1.20.8−5.6 | 2.4-4.8 | mRS=-2 |
SP | 0.8−3.6-0.6−0.8 | -2.8-1.4 | mSP=0.5 |
From the slopes above, we can see that PQ and RS have equal slopes. Thus, they are parallel. The same can be said for QR and SP. To determine if QR⊥RS, we can see if the product of their slopes equals -1. mQR⋅mRS=0.5(-2)=-1 Thus, QR⊥RS. It follows that all pairs of adjacent sides are perpendicular. Therefore, the quadrialteral is a rectangle.