Topic 6 of 7

Coordinate geometry

Step 12 · Geometry topics

Use the distance and midpoint formulas, the slopes of parallel and perpendicular lines, and the equation of a circle to solve geometry problems with coordinates.

Distance and midpoint

On the coordinate plane, the distance between two points comes from the Pythagorean theorem, and the midpoint is the average of their coordinates.

d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}} \qquad M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Example: Distance and midpoint

Find the distance and the midpoint between (−1, 2) and (5, 10).

Distance: √(6² + 8²) = √100 = 10.
Midpoint: the averages of −1 and 5, and of 2 and 10: (2, 6).

AnswerDistance 10, midpoint (2, 6)

Example: An endpoint from a midpoint

M(3, 4) is the midpoint of AB, and A is (1, −2). Find B.

B is as far past M as A is before it.
x goes 1, 3, 5 and y goes −2, 4, 10.

AnswerB(5, 10)

M is halfway: the step from A to M is the same as the step from M to B.

Slopes of parallel and perpendicular lines

Parallel lines have equal slopes. Perpendicular lines have slopes that multiply to −1: each is the negative reciprocal of the other.

Slope of a line	Parallel slope	Perpendicular slope
2	2	$-\frac{1}{2}$
$-\frac{3}{4}$	$-\frac{3}{4}$	$\frac{4}{3}$
0 (flat)	0	undefined (vertical)

Example: A parallel line

Write the equation of the line through (2, 7) parallel to y = 3x − 1.

The same slope: 3.
y − 7 = 3(x − 2), so y = 3x + 1.

Answery = 3x + 1

Example: A perpendicular line

Write the equation of the line through (4, 1) perpendicular to y = 2x + 3.

The perpendicular slope is $-\frac{1}{2}$ .
$y - 1 = -\frac{1}{2}(x - 4)$ .

Answer $y = -\frac{1}{2}x + 3$

Example: A right angle?

Is the triangle with corners (0, 0), (4, 2) and (3, −6) a right triangle?

Slope from (0, 0) to (4, 2): 2 ÷ 4 = 0.5.
Slope from (0, 0) to (3, −6): −6 ÷ 3 = −2.
0.5 times −2 is −1.

AnswerYes: the sides meeting at (0, 0) are perpendicular

Equation of a circle

A circle is every point at a fixed distance r from its center (h, k). The distance formula turns that into its equation.

(x - h)^{2} + (y - k)^{2} = r^{2}

The circle (x − 2)² + (y − 1)² = 9: center (2, 1) and radius 3.

Example: Reading the equation

Find the center and radius of (x + 4)² + (y − 3)² = 25.

x + 4 is x − (−4), so h = −4, and k = 3.
r² = 25, so r = 5.

AnswerCenter (−4, 3), radius 5

Example: Completing the square

Find the center and radius of x² + y² − 6x + 8y = 0.

Complete each square: (x² − 6x + 9) + (y² + 8y + 16) = 9 + 16.
So (x − 3)² + (y + 4)² = 25.

AnswerCenter (3, −4), radius 5

Try it yourself

Work each one on paper first, then open the answer to check.

Find the distance between (2, 3) and (8, 11).
Show answer
10
Find the midpoint of (−4, 6) and (8, −2).
Show answer
(2, 2)
What is the slope of a line perpendicular to $y = \frac{2}{3}x + 1$ ?
Show answer
$-\frac{3}{2}$
Write the equation of the line through (0, 4) parallel to y = −2x + 7.
Show answer
y = −2x + 4
Write the equation of the circle with center (1, −2) and radius 6.
Show answer
(x − 1)² + (y + 2)² = 36
Find the center and radius of x² + (y − 5)² = 49.
Show answer
Center (0, 5), radius 7