In the standard (x, y) coordinate plane, line L_1 passes through the points (1, 3) and (4,… — ACT

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(1, 3)$ and $(4, 8)$ . Line $L_2$ is perpendicular to line $L_1$ . If line $L_2$ passes through the points $(5, k)$ and $(10, 1)$ , what is the value of $k$ ?

Answer: 4

Answer

The value of

k

4

The correct answer is

4

. First, determine the slope of line

L_1

using the points

(1, 3)

and

(4, 8)

, which is

\frac{8 - 3}{4 - 1} = \frac{5}{3}

. Because line

L_2

is perpendicular to

L_1

, its slope must be the negative reciprocal of

\frac{5}{3}

, which is

-\frac{3}{5}

. Next, set up the slope equation for

L_2

with the points

(5, k)

and

(10, 1)

, yielding

\frac{1 - k}{10 - 5} = -\frac{3}{5}

. Simplifying the equation gives

\frac{1 - k}{5} = -\frac{3}{5}

, which reduces to

1 - k = -3

. Solving for

k

gives

k = 4

Step-by-Step Solution

Calculate the slope of line

L_1

using the coordinates of the two given points,

(1, 3)

and

(4, 8)

The slope of line

L_1

\frac{5}{3}

Using the slope formula

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

, we find

m_1 = \frac{8 - 3}{4 - 1} = \frac{5}{3}

Determine the slope of line

L_2

based on the perpendicular relationship between

L_1

and

L_2

The slope of line

L_2

-\frac{3}{5}

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

\frac{5}{3}

-\frac{3}{5}

Use the coordinates

(5, k)

and

(10, 1)

on line

L_2

to write an expression for its slope, set it equal to

-\frac{3}{5}

, and solve for

k

k = 4

The slope expression is

\frac{1 - k}{10 - 5} = \frac{1 - k}{5}

. Setting this equal to the perpendicular slope gives

\frac{1 - k}{5} = -\frac{3}{5}

. Multiplying both sides by

5

results in

1 - k = -3

. Adding

k

to both sides and adding

3

to both sides yields

k = 4

Key Concept

The slopes of perpendicular lines in a coordinate plane are negative reciprocals of each other, meaning their product is

-1

(

m_1 \cdot m_2 = -1

Estimated Time:1m 15s

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