Question 398·Medium·Algebra

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A map uses an $xy$ -coordinate grid. In the graph shown, a straight river is represented by line $t$ , and a rescue boat is located at point $R$ .

The boat will travel along a straight path that passes through $R$ and is perpendicular to line $t$ . Which choice gives an equation of the boat’s path?

A graph showing a line, a separate point, and the word perpendicular calls for two facts about a new line: its slope and a point it passes through. Read two exact points on the given line and type their rise-over-run fraction in Desmos. Then take the negative reciprocal and use the separate point to find the intercept. Keeping the original slope won't give a perpendicular path.

Hints

Hint 1
To find slope, divide the change in $y$ by the matching change in $x$ . Use the two dots on line $t$ , not the separate boat point. Does the river rise or fall as you move right?
Hint 2
Perpendicular lines meet at a right angle. For nonvertical lines, their slopes multiply to $-1$ , so flip the slope fraction and change its sign. What slope makes that product $-1$ ?
Hint 3
The $y$ -intercept is the value of $y$ when $x=0$ , not the $y$ -coordinate of any point. Put the boat point into $y=mx+b$ to find $b$ .

Step-by-step

Find the perpendicular slope, then use the boat’s point
Step 1
Find the river’s slope
The marked dots $(-4,6)$ and $(6,1)$ both sit on line $t$ . The slope is the change in $y$ divided by the change in $x$ , using the same point order for both. Type $\frac{1-6}{6-(-4)}$ in Desmos; it shows $-0.5$ , or $-\frac12$ with the fraction button. Don't use $R$ to measure the river's slope.
Step 2
Turn the river’s slope into the boat’s slope
A perpendicular path meets the river at a right angle. For nonvertical lines, take the negative reciprocal of the river's slope: flip the fraction and change its sign. The river goes right $2$ , down $1$ ; the boat's right-angle direction goes right $1$ , up $2$ . So its slope is $2$ , not $-\frac12$ .
Step 3
Put the boat’s point into a line equation
In slope-intercept form, $y=mx+b$ , $m$ is the slope and $b$ is the value of $y$ at $x=0$ . The boat's slope is $2$ , and it passes through $R=(2,-3)$ , so put that point into $y=2x+b$ : $-3=2(2)+b$ . The $-3$ isn't $b$ because $R$ is at $x=2$ , not $0$ .
Step 4
Find the intercept
Type $-3\sim2(2)+b$ in Desmos. The $\sim$ tells Desmos to find the value of $b$ that makes both sides match. Under PARAMETERS, it shows $b=-7$ , so the boat's $y$ -intercept is $-7$ .
Step 5
Write the boat’s path
Put slope $2$ and intercept $-7$ into slope-intercept form: $y=mx+b$ becomes $y=2x-7$ , the rescue boat's perpendicular path through $R$ . Choice A.

Question 398·Medium·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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