A line segment is bisected by a line with the equation 3 y + 5 x = 2 . If one end of the line segment is at ( 1 , 4 ), where is the other end?

1 Answer
Douglas K.
Nov 4, 2016

The other end of the line is at (-116/34, 46/34)

Explanation:

Write the given line in slope-intercept form:

y = -5/3x + 2/3 [1]

The slope, m = -5/3

Because the bisected line is perpendicular, we know that its slope, n, is the negative reciprocal of the bisector:

n = -1/m

n = 3/5

Use the given point, (1, 4) and the slope 3/5 to find the value of b in the slope-intercept form y = mx + b:

4 = 3/5(1) + b

b = 4 - 3/5

b = 17/5

The equation of the bisected line is:

y = 3/5x + 17/5 [2]

Find the x coordinate of the intersection by subtracting equation [1] from equation [2]:

y - y = 3/5x + 5/3x + 17/5 - 2/3

0 =34/15x + 41/15

0 =34x + 41

-34x = 41

x = -41/34

Deltax = -41/34 - 1 = -75/34

The x coordinate of the other end of the line is:

1 + 2Deltax = 1 + 2(-75/34) = -116/34

To find the corresponding y coordinate, substitute -116/34 for x in equation 2:

y = 3/5(-116/34) + 17/5

y = 46/34

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