In the xy-plane, point M(3, 3) is the midpoint of line segment AB, where point A has coord… — GRE General Test

In the $xy$ -plane, point $M(3, 3)$ is the midpoint of line segment $AB$ , where point $A$ has coordinates $(1, 2)$ . Line $L$ is perpendicular to line segment $AB$ and passes through point $B$ . What is the $x$ -intercept of line $L$ ?

Answer: 7

Answer

Using the midpoint formula with

A(1, 2)

and

M(3, 3)

gives endpoint

B(5, 4)

. The slope of segment

AB

\frac{4-2}{5-1} = \frac{1}{2}

, so line

L

, being perpendicular to

AB

, has slope

-2

. The equation of line

L

passing through

(5, 4)

y - 4 = -2(x - 5)

, which simplifies to

y = -2x + 14

. Setting

y = 0

gives

0 = -2x + 14

, so the

x

-intercept is

7

Step-by-Step Solution

Calculate the coordinates of endpoint B using the midpoint formula.

Point B has coordinates

(5, 4)

Midpoint

M(x_m, y_m) = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right)

. Solving

\frac{1 + x_B}{2} = 3

yields

x_B = 5

, and solving

\frac{2 + y_B}{2} = 3

yields

y_B = 4

Determine the slope of segment AB.

The slope

m_{AB} = \frac{1}{2}

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

between

(1, 2)

and

(5, 4)

gives

\frac{4 - 2}{5 - 1} = \frac{2}{4} = \frac{1}{2}

Determine the slope of line L.

The slope of line L is

-2

Perpendicular lines have negative reciprocal slopes. The negative reciprocal of

\frac{1}{2}

-2

Find the equation of line L and solve for its x-intercept.

The x-intercept is 7.

Using point-slope form with

B(5, 4)

and

m = -2

y - 4 = -2(x - 5) \implies y = -2x + 14

. Setting

y = 0

gives

0 = -2x + 14 \implies x = 7

Key Concept

Midpoint Formula, Perpendicular Slopes, and Line Intercepts

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