7.1 Coordinate geometry: slope, distance, midpoint & graphing lines

Coordinate Geometry on the PERT

The coordinate plane is built from two number lines that cross at right angles. The horizontal line is the x-axis, the vertical line is the y-axis, and they meet at the origin, written (0, 0). Every point is named by an ordered pair (x, y): the first number tells you how far to move left or right, and the second tells you how far to move up or down. The PERT Mathematics subtest expects you to read points, compute slope, write line equations, and use the distance and midpoint formulas. This section walks through each skill with fully worked arithmetic.

Slope: rise over run

Slope measures steepness. For two points (x1, y1) and (x2, y2), the formula is:

m = (y2 - y1) / (x2 - x1)

The top (rise) is the change in y; the bottom (run) is the change in x. Keep the points in the same order top and bottom.

Worked example. Find the slope through (2, 3) and (6, 11). Let (x1, y1) = (2, 3) and (x2, y2) = (6, 11).

m = (11 - 3) / (6 - 2) = 8 / 4 = 2.

The slope is 2, meaning the line rises 2 units for every 1 unit right. A positive slope rises left-to-right; a negative slope falls; a horizontal line has slope 0; a vertical line has an undefined slope, because its run is 0 and you cannot divide by 0.

Slope-intercept form: y = mx + b

The most useful line equation is y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

Worked example. A line has slope 3 and crosses the y-axis at (0, -4). Its equation is y = 3x - 4. To find y when x = 5: y = 3(5) - 4 = 15 - 4 = 11, so the point (5, 11) is on the line.

Point-slope form

When you know the slope and one point (x1, y1), use point-slope form:

y - y1 = m(x - x1)

Worked example. Write the line with slope 4 through (1, 2).

y - 2 = 4(x - 1). Distribute the 4: y - 2 = 4x - 4. Add 2 to both sides: y = 4x - 2. In slope-intercept form the slope is 4 and the y-intercept is -2.

Parallel and perpendicular lines

Two lines are parallel when they have equal slopes and never meet. Two lines are perpendicular when they cross at a right angle; their slopes are negative reciprocals, meaning the two slopes multiply to -1.

Worked example. A line has slope 2/3. A parallel line also has slope 2/3. A perpendicular line has slope -3/2: flip 2/3 to 3/2, then change the sign. Check: (2/3)(-3/2) = -1, so the lines really are perpendicular.

The distance formula

To find the distance between two points, use the distance formula, which comes straight from the Pythagorean theorem:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Worked example. Find the distance from (1, 2) to (4, 6).

Horizontal change: 4 - 1 = 3. Vertical change: 6 - 2 = 4.

d = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.

The distance is exactly 5 units. Squaring removes any negative sign, so the order in which you list the points does not change the result.

The midpoint formula

The midpoint is the point halfway between two endpoints. Average the x-values and average the y-values:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Worked example. Find the midpoint of (2, 3) and (8, 7).

x: (2 + 8)/2 = 10/2 = 5. y: (3 + 7)/2 = 10/2 = 5. The midpoint is (5, 5).

Writing a line from two points

Some questions give you two points and ask for the equation. First find the slope, then plug into point-slope form, then simplify.

Worked example. Write the equation of the line through (2, 1) and (4, 7).

Slope: m = (7 - 1)/(4 - 2) = 6/2 = 3. Now use point-slope form with the point (2, 1): y - 1 = 3(x - 2). Distribute: y - 1 = 3x - 6. Add 1 to both sides: y = 3x - 5. The y-intercept is -5, so the line crosses the y-axis at (0, -5). You would reach the same answer using the other point (4, 7), which is a handy way to check your work.

Graphing a line

To graph y = mx + b, plot the y-intercept first, then use the slope as rise-over-run to find a second point.

Worked example. Graph y = 2x + 1. Start at the y-intercept (0, 1). The slope 2 = 2/1 means "up 2, right 1," which gives the point (1, 3). Plot (0, 1) and (1, 3), then draw the straight line through them. To check a third point, substitute: at x = 3, y = 2(3) + 1 = 7, so (3, 7) also lies on the line. Reading these values back from a graph, including the intercept, a second point, and the direction of the slope, is exactly what PERT questions ask you to do.