Perpendicular Bisector

Algebra Level 3

Consider the line segment whose endpoints are the x x - and y y -intercepts of 4 x + y − 24 = 0 4 x + y -24 = 0 , and let line l l be the perpendicular bisector of this segment. If line l l passes through ( 11 , a ) (11,\ a) , what is the value of a a ?

11 11 13 13 12 12 14 14

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Mahindra Jain by Mahindra Jain

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2 solutions

Paola Ramírez
May 16, 2015

Write the linear equation in slope-intercept form:

y = 24 − 4 x \boxed{y=24-4x}

m = − 4 ∴ m=-4 \therefore the perpendicular bisector have m = 1 4 m=\frac{1}{4}

Then calculate intersections with axis -x and -y which are: ( 0 , 24 ) (0,24) and ( 6 , 0 ) (6,0)

The midpoint of this segment is ( 6 + 0 2 , 24 + 0 2 ) = ( 3 , 12 ) (\frac{6+0}{2},\frac{24+0}{2})=(3,12)

Finally, find a ⇒ 12 − a 3 − 11 = 1 4 ⇒ a = 14 a \Rightarrow \frac{12-a}{3-11}=\frac{1}{4} \Rightarrow \boxed{a=14}

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Sim Jun
Mar 4, 2014

X-intercept= 24/4=6 Y-intercept= 24 Perpendicular bisector= (3,12) Gradient of line l= -1/-4=1/4 A= 12+ (11-3)(1/4) = 14

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