If two lines are perpendicular the slope of one is the negative reciprocal of

# If two lines are perpendicular the slope of one is

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. m. . Theorem 9.10bGivenLinesl1and l2with slope ProveProofChoose any point on m is k . EXAMPLE 1The vertices of quadrilateral ABCDare A(2,4),B(6,2),C(2, 6), and D(1, 2).a.Show that two sides of the quadrilateral are parallel. b.Show that the quadrilateral has two right angles. 21ml1l2If the slopes of two non-vertical lines in the coordinate plane are equal, thenthe lines are parallel. Parallel Lines in the Coordinate Plane343 l1kl2xy
Solution The slope of . The slope of . The slope of . The slope of . a. and are parallel because they have equal slopes. b. The slope of is the negative reciprocal of the slope of , so they are perpendicular. Therefore, B is a right angle. The slope of is the negative reciprocal of the slope of , so they are perpendicular. Therefore, A is a right angle. Answers a. b. A and B are right angles. EXAMPLE 2 Write an equation for l 1 , the line through ( 2, 5) that is parallel to the line l 2 whose equation is 2 x y 7. Solution (1) Solve the equation of l 2 for y : (2) Find the slope of l 2 . The slope of a line in slope-intercept form is the coefficient of x : (3) Find the slope of l 1 , which is equal to the slope of l 2 : (4) Use the definition of slope to write an equation of l 1 . Let ( x, y ) and ( 2, 5) be two points on l 1 : Answer y 2 x 1 or 2 x y 1 DA BC DA AB BC AB DA BC DA 5 2 4 2 2 2 2 ( 2 1 ) 5 2 6 3 5 2 2 CD 5 2 2 6 2 1 2 2 5 2 4 2 3 5 4 3 BC 5 6 2 ( 2 2 ) 2 2 6 5 8 2 4 5 2 2 AB 5 2 2 2 ( 2 4 ) 6 2 2 5 2 4 5 1 2 344 Parallel Lines O C B A D 1 y x 1 2 x y 7 y 2 x 7 y 2 x 7 slope of l 1 2 slope 2 y 5 2( x 2) y 5 2 x 4 y 2 x 1 y 2 5 x 2 ( 2 2 ) y 1 2 y 2 x 1 2 x 2 slope