A line, L_1, is perpendicular to a second line whose equation is 5x - 2y = 15. The line L_… — ACT

A line, $L_1$ , is perpendicular to a second line whose equation is $5x - 2y = 15$ . The line $L_1$ intersects the $y$ -axis at $(0, 12)$ and passes through the midpoint of a line segment with endpoints at $(3, k)$ and $(7, 6)$ in the standard $(x, y)$ coordinate plane. What is the value of $k$ ?

Answer: 14

Answer

The correct value of

k

is 14.

The slope of the line

5x - 2y = 15

is found by solving for

y

, yielding

y = \frac{5}{2}x - 7.5

. The slope of any perpendicular line is the negative reciprocal of

\frac{5}{2}

, which is

-\frac{2}{5}

. Given the

y

-intercept

(0, 12)

, the equation of the perpendicular line

L_1

y = -\frac{2}{5}x + 12

. The midpoint of the segment with endpoints

(3, k)

and

(7, 6)

is calculated as

\left(\frac{3+7}{2}, \frac{k+6}{2}\right) = \left(5, \frac{k+6}{2}\right)

. Since the midpoint lies on

L_1

, substituting

x = 5

into the line equation gives

y = -\frac{2}{5}(5) + 12 = 10

. Equating this to the midpoint's

y

-coordinate expression gives

\frac{k+6}{2} = 10

, which solves to

k = 14

Step-by-Step Solution

Find the slope of the line

5x - 2y = 15

by converting it to slope-intercept form (

y = mx + b

The slope of the line is

\frac{5}{2}

Rewriting the equation as

-2y = -5x + 15

and dividing by

-2

isolates

y

and reveals the slope.

Determine the perpendicular slope for line

L_1

The slope of

L_1

-\frac{2}{5}

Perpendicular lines have slopes that are negative reciprocals of one another.

Formulate the equation of line

L_1

using its slope and the given

y

-intercept

(0, 12)

The equation of

L_1

y = -\frac{2}{5}x + 12

The slope-intercept form is

y = mx + b

, where

m

is the slope and

b

is the

y

-coordinate of the

y

-intercept.

Find the midpoint of the line segment with endpoints

(3, k)

and

(7, 6)

in terms of

k

The midpoint is

\left(5, \frac{k+6}{2}\right)

The midpoint formula calculates the average of the

x

-coordinates and the average of the

y

-coordinates.

Substitute the midpoint coordinates into the equation of

L_1

and solve for

k

k = 14

Since the midpoint lies on line

L_1

, substituting its

x

and

y

values into the equation must satisfy the equality.

Key Concept

Using perpendicular slopes and the midpoint formula to determine unknown coordinate values.

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