Coordinate Geometry and Slope

Coordinate geometry puts algebra and geometry on the same page. Almost every question here is about a line, and almost every line question is about its slope.

Plotting

A point is written (x, y): across first, then up. The order is the most common first-week error and the test still asks it.

The axes split the plane into four quadrants, numbered counterclockwise from the top right.

So (-3, 5) is in quadrant II, and (4, -2) is in quadrant IV. A question asking which quadrant a point falls in is asking only for the pair of signs.

Slope

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Slope = rise / run = the change in y / the change in x.

For (1, 2) and (5, 5): the rise is 5 minus 2, which is 3, and the run is 5 minus 1, which is 4. Slope is 3/4.

Subtract in the same order in both places. Doing 5 minus 2 on top and 1 minus 5 on the bottom gives -3/4, which is the wrong sign and is a choice. Pick a point to call "first" and be consistent.

Reading a slope's sign and size

Horizontal is zero, vertical is undefined, and they are frequently swapped. A horizontal line rises 0 over any run, so 0 divided by something is 0. A vertical line has a run of 0, and dividing by 0 is not a number.

Steeper means a larger absolute value. A slope of 5 is steeper than a slope of 1/2, and a slope of -5 is just as steep as 5 while going the other way.

Parallel and perpendicular

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are negative reciprocals: flip the fraction and change the sign.

A line of slope 2/3 is perpendicular to a line of slope -3/2. Their product is -1, which is the check.

The equation of a line

y = mx + b, where m is the slope and b is the y-intercept.

y = -2x + 7

The slope is -2 and the line crosses the y-axis at 7.

To find where a line crosses an axis:

• y-intercept: set x to 0. Here y = 7.
• x-intercept: set y to 0. Here 0 = -2x + 7, so x = 3.5.

Watch for equations not yet in that form. In 2y = 6x - 8, divide through by 2 to get y = 3x - 4, and only then read the slope as 3. Reading the coefficient before isolating y is the standard error, and it gives 6 here.

Distance and midpoint

Midpoint: average the x values, average the y values.

For (2, 3) and (8, 11): the midpoint is ((2 + 8)/2, (3 + 11)/2), which is (5, 7).

Distance: the Pythagorean theorem. The horizontal gap is one leg, the vertical gap is the other, and the distance between the points is the hypotenuse.

For the same two points: the horizontal gap is 6, the vertical gap is 8, so the distance is the square root of 36 plus 64, which is the root of 100, or 10.

There is no need to memorize the distance formula as a formula. Draw the right triangle, count the legs, apply the theorem. That is what the formula is, and the triangle version is harder to misremember.

What you can skip

Across the 31 questions on this topic:

• Point-slope and standard form conversions never appear by name in this topic; slope-intercept form carries the line questions.
• The distance formula as a formula. The distance questions are the Pythagorean theorem on a grid; draw the right triangle.

Where people lose points

Reversing the order in the slope subtraction, giving the wrong sign.

Swapping rise and run, giving the reciprocal. Rise is on top, and the word comes first in the phrase.

Swapping zero and undefined for horizontal and vertical lines.

Reading the slope off an equation that is not solved for y.

Reversing x and y when plotting.

Using the distance formula when the midpoint was asked, or the reverse.

Work one in under a minute

What is the slope of the line through (-2, 5) and (4, -7)?

Take (-2, 5) as first.

Rise: -7 minus 5 is -12. Run: 4 minus (-2) is 6.

Slope: -12 / 6 is -2.

Check the sign against the picture: as x goes from -2 to 4 the line drops from 5 to -7, so it falls left to right and the slope must be negative. It is.

Where this leads

A system of two equations is two lines meeting, and the distance formula is the Pythagorean theorem in coordinates.