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 $k$ ?

Cevap: 14

Cevap

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

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru