In the xy-plane, line L passes through the points (-2, 5) and (4, 1). Line M is the perpen… — GRE General Test

In the $xy$ -plane, line $L$ passes through the points $(-2, 5)$ and $(4, 1)$ . Line $M$ is the perpendicular bisector of the line segment connecting these two points. What is the $y$ -intercept of line $M$ ?

A
$-\frac{3}{2}$
B
$\frac{1}{2}$
$\frac{3}{2}$ Answer
D
$\frac{7}{3}$
E
$\frac{11}{3}$

Answer

\frac{3}{2}

The midpoint of the segment with endpoints

(-2, 5)

and

(4, 1)

\left(\frac{-2+4}{2}, \frac{5+1}{2}\right) = (1, 3)

. The slope of the segment is

\frac{1-5}{4-(-2)} = -\frac{2}{3}

. Therefore, the perpendicular bisector (line

M

) has a slope equal to the negative reciprocal,

\frac{3}{2}

. Using point-slope form with point

(1, 3)

, line

M

has the equation

y - 3 = \frac{3}{2}(x - 1)

, which simplifies to

y = \frac{3}{2}x + \frac{3}{2}

. The

y

-intercept is

\frac{3}{2}

Step-by-Step Solution

Calculate the midpoint of the line segment with endpoints

(-2, 5)

and

(4, 1)

Midpoint

(x_m, y_m) = \left(\frac{-2 + 4}{2}, \frac{5 + 1}{2}\right) = (1, 3)

The perpendicular bisector passes through the midpoint of the segment.

Determine the slope of line

L

m_L = \frac{1 - 5}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}

Slope is calculated as change in

y

divided by change in

x

Determine the slope of line

M

(perpendicular bisector)

m_M = -\frac{1}{m_L} = \frac{3}{2}

Perpendicular lines have slopes that are negative reciprocals of each other.

Find the equation of line

M

using point-slope form through

(1, 3)

and evaluate its

y

-intercept

y - 3 = \frac{3}{2}(x - 1) \implies y = \frac{3}{2}x + \frac{3}{2}

. Setting

x = 0

gives

y = \frac{3}{2}

The

y

-intercept is the value of

y

when

x = 0

Key Concept

Perpendicular Bisectors and Slope-Intercept Form

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