In the xy-plane, the graph of a linear function f has a y-intercept of (0, r) and an x-int… — SAT

In the $xy$ -plane, the graph of a linear function $f$ has a $y$ -intercept of $(0, r)$ and an $x$ -intercept of $(s, 0)$ , where $r$ and $s$ are positive constants. The line $y = -2x$ is perpendicular to the line that passes through the origin $(0,0)$ and the midpoint of the segment connecting the two intercepts of $f$ . If $r = 12$ , what is the value of $s$ ?

Answer: 24

Answer

The midpoint of the segment connecting

(0, 12)

and

(s, 0)

(\frac{s}{2}, 6)

. The line passing through the origin

(0, 0)

and this midpoint has a slope of

\frac{6}{s/2} = \frac{12}{s}

. Since this line is perpendicular to the line

y = -2x

, which has a slope of

-2

, its slope must be the negative reciprocal of

-2

, which is

\frac{1}{2}

. Equating these two slopes, we get

\frac{12}{s} = \frac{1}{2}

, which simplifies to

s = 24

Step-by-Step Solution

Find the coordinates of the intercepts and their midpoint.

The intercepts are

(0, 12)

and

(s, 0)

, and their midpoint is

(\frac{s}{2}, 6)

The intercepts of the function

f

form a line segment whose midpoint must be calculated.

Calculate the slope of the line passing through the origin and the midpoint.

The slope is

\frac{12}{s}

A line passing through the origin

(0,0)

and a point

(x_1, y_1)

has a slope of

\frac{y_1}{x_1}

Relate the slope of the line to the perpendicular line

y = -2x

The slope of the line must be

\frac{1}{2}

Perpendicular lines have slopes that are negative reciprocals of each other, and the negative reciprocal of

-2

\frac{1}{2}

Solve for the value of

s

s = 24

Setting the two expressions for the slope equal,

\frac{12}{s} = \frac{1}{2}

, yields

s = 24

Key Concept

Linear Functions and Graphs

Estimated Time:2m 30s

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