In the xy-plane, line k is defined by the equation 2x - y = 8. Line m is perpendicular to … — GRE General Test

In the $xy$ -plane, line $k$ is defined by the equation $2x - y = 8$ . Line $m$ is perpendicular to line $k$ and passes through the points $(-1, 9)$ and $(t, 5)$ . What is the value of $t$ ?

A
$-9$
B
$-3$
C
$1$
$7$ Answer
E
$9$

Answer

The value of

t

7

The line

k

has equation

y = 2x - 8

, giving a slope of

2

. A line perpendicular to it must have a slope of

-\frac{1}{2}

. Using the slope formula for line

m

passing through

(-1, 9)

and

(t, 5)

yields

\frac{5 - 9}{t - (-1)} = -\frac{1}{2}

, which simplifies to

\frac{-4}{t + 1} = -\frac{1}{2}

, leading directly to

t = 7

Step-by-Step Solution

Find the slope of line

k

Converting

2x - y = 8

to slope-intercept form gives

y = 2x - 8

, so the slope of line

k

m_k = 2

The slope-intercept form

y = mx + b

directly identifies the slope

m

of a line.

Determine the slope of line

m

Since line

m

is perpendicular to line

k

, its slope is

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

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

Set up the slope formula for line

m

using the given points and solve for

t

\frac{5 - 9}{t - (-1)} = -\frac{1}{2} \implies \frac{-4}{t + 1} = -\frac{1}{2} \implies \frac{4}{t + 1} = \frac{1}{2} \implies t + 1 = 8 \implies t = 7

The slope between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

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

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

Estimated Time:1m 30s

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