# Coordinate Geometry

## Geometry: Coordinate Geometry

Approach every question the same way to minimize mistakes.  Start by reading the question carefully and identifying the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches your bottom line.

In the xy-plane, line l passes through the points (0, 0) and (2, 5). Line m is perpendicular to line l. What is the slope of line m?

Bottom line: slope m = ?

Assess your Options:  You could draw out a graph and solve this visually, but that is a waste of time if you know the formula to find the slope of a line.

Attack the Problem:  You are given the most information about line l, so start with that line.  You should have the formula for slope memorized:

$slope=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

It is easiest just to think about slope as the change in y-values over the change in x-values.  If you look up at the original points that you have been given, from zero the y-values go up to 5 and the x-values go up to 2.  You now have 5 over 2.

The slope of line l is $\frac{5}{2}$.

At this point, some students will think they are finished and select answer (D).  However, your bottom line was the slope of line m!  The problem tells you that line m is perpendicular to line l.  In order to find a perpendicular line, you must take the opposite reciprocal of the first line; in essence you must flip the sign (negative or positive) and the numbers (a fraction or whole number).

The slope of line m is $-\frac{2}{5}$.

(A) $-\frac{5}{2}$
(B) $-\frac{2}{5}$
(C) $\frac{2}{5}$
(D) $\frac{5}{2}$
(E) 5

On sat.collegeboard.org, 47% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

# Coordinate Geometry

## Geometry: Coordinate Geometry

Read the following SAT test question and then select the correct answer.

Always be sure to read the question carefully and make a note of the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

In the xy-plane, line l passes through the points (a, 0) and (0, 2a), where a > 1.  What is the slope of line l?

Bottom Line: slope of l = ?

Assess Your Options: You could select a number larger than 1, plug it in for the variable a, and then work the problem.  However, if you peek down at the answer choices, notice that some have a variable still in the problem.  It will take you longer to plug in a number than to work the problem using the variables.

Attack the Problem: Your bottom line is a slope, so use the formula for the slope of a line. The formula for slope of a line is:

$\frac{rise}{run}\: or\: \frac{\Delta y}{\Delta x}$

To find the change in y coordinates, subtract the first y-value from the second y-value.  Do the same with the x values:

$\frac{2a-0}{0-a}=\frac{2a}{-a}=-2$

The variable will cancel when you simplify the problem.  Your answer is -2.

Loop Back:  You found the slope of the line, so you are ready to look down at your answer choices.

(A) -2
(B)$-\frac{1}{2}$
(C) 2
(D) -2a
(E) 2a

On sat.collegeboard.org, 42% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

# Calculating the Slope of a Line

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## 6/8 Calculating the slope of a Line

In the xy-plane, line  passes through the points and . Line  is perpendicular to line . What is the slope of line ?

The first step here is to identify the bottom line. In this case we are looking for the slope of a line that is perpendicular to the line that passes through the points (0,0) and (2,5). The first step to answering this problem is to calculate the slope of the line that passes through the points given. Remember that the slope of a line is equal to the rise over the run so you have

It is essential that you don't forget to answer the bottom line for this question. Although this is a fairly easy question, it is very easy to make a sloppy mistake and miss some easy points. We are looking for the slope of a line that is perpendicular to the line that passes through the points given. The slope of a  perpendicular line is always the negative reciprocal of the slope of the line it is perpendicular to. So we take the original slope we calculated, negate it, and flip it.

Now we simply look at the answers and choose the answer that matches our prediction.

(A)
(B)
(C)
(D)
(E)

As you can see, answer B matches our prediction exactly. Note that if we had failed to answer the bottom line and only calculated the slope of the line that went through the points, we would have picked answer (D). Even though this was a fairly easy problem, more than 50% of the responses were incorrect!

On sat.collegeboard.com 49% of the responses were correct.

For more help with math, visit www.myknowsys.com!