In the xy-plane, line k passes through the points (1, -2) and (5, 6). Line m is the perpen… — GRE General Test

In the $xy$ -plane, line $k$ passes through the points $(1, -2)$ and $(5, 6)$ . Line $m$ is the perpendicular bisector of the line segment connecting $(1, -2)$ and $(5, 6)$ . Which of the following statements must be true? Select all that apply.

Line $m$ passes through the point $(7, 0)$ .Answer
Line $m$ has a $y$ -intercept of $(0, 3.5)$ .Answer
The distance from the origin $(0, 0)$ to the midpoint of the line segment is $\sqrt{13}$ .Answer
D
The slope of line $m$ is $2$ .
E
Line $m$ is parallel to the line with equation $2x - y = 4$ .

Answer

The statements asserting that line

m

passes through

(7, 0)

, that line

m

has a

y

-intercept of

(0, 3.5)

, and that the distance from the origin to the midpoint of the segment is

\sqrt{13}

are all true.

Line

k

has slope

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

and segment midpoint

(3, 2)

. Line

m

, as the perpendicular bisector, has slope

-\frac{1}{2}

and equation

y = -\frac{1}{2}x + 3.5

. Substituting

x = 7

gives

y = 0

, so

(7, 0)

lies on line

m

. Substituting

x = 0

gives

y = 3.5

, confirming the

y

-intercept. Finally, the distance from

(0,0)

to the midpoint

(3,2)

\sqrt{3^2 + 2^2} = \sqrt{13}

. Thus, all three of these statements are correct.

Step-by-Step Solution

Find the midpoint and slope of the segment connecting

(1, -2)

and

(5, 6)

Midpoint

M = \left(\frac{1+5}{2}, \frac{-2+6}{2}\right) = (3, 2)

. Slope of segment

m_k = \frac{6 - (-2)}{5 - 1} = \frac{8}{4} = 2

Line

m

is the perpendicular bisector, so it passes through midpoint

M

and has a slope perpendicular to

m_k

Determine the slope and equation of line

m

Perpendicular slope

m_m = -\frac{1}{m_k} = -\frac{1}{2}

. Point-slope equation through

(3, 2)

y - 2 = -\frac{1}{2}(x - 3) \implies y = -\frac{1}{2}x + \frac{7}{2}

Perpendicular lines have negative reciprocal slopes.

Evaluate each given statement against the calculated values.

Line

m

contains

(7, 0)

because

0 = -\frac{1}{2}(7) + \frac{7}{2}

. The

y

-intercept is

(0, 3.5)

. Distance from

(0,0)

(3,2)

\sqrt{3^2 + 2^2} = \sqrt{13}

. Statements claiming slope is

2

or that line

m

is parallel to

2x - y = 4

are incorrect.

Determines which statements satisfy all mathematical conditions.

Key Concept

Perpendicular Bisectors, Slopes of Perpendicular Lines, and Distance Formula

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