UNIT 9 • GEOMETRY & TRIGONOMETRY

Coordinate Geometry
Points, Lines & Slopes

Coordinate geometry turns points and lines into numbers: the distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ , and the slope of the line through them is $\frac{y_2 - y_1}{x_2 - x_1}$ .

THE CORE FORMULAS8 TOOLS

\begin{gathered} d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ M = \left(\tfrac{x_1 + x_2}{2}, \tfrac{y_1 + y_2}{2}\right) \\ m = \frac{y_2 - y_1}{x_2 - x_1} \end{gathered}

1. Label the PointsCall them

(x_1, y_1)

and

(x_2, y_2)

and keep that order all the way through.

2. Watch the SignsSubtracting a negative adds:

3 - (-2) = 5

. Brackets help.

3. Draw ItA quick sketch shows whether the answer is sensible: a positive slope rises to the right.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

For $A(1, 2)$ and $B(3, 8)$ , find the distance AB, the midpoint of AB, and the equation of line AB.

STEP 01 • DISTANCE

AB = \sqrt{(3 - 1)^2 + (8 - 2)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \approx 6.325

STEP 02 • MIDPOINT

M = \left(\frac{1 + 3}{2}, \frac{2 + 8}{2}\right) = (2, 5)

STEP 03 • SLOPE AND EQUATION

m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3

y - 2 = 3(x - 1) \;\Rightarrow\; y = 3x - 1 \;\Rightarrow\; 3x - y - 1 = 0

Check with B: $3(3) - 1 = 8$ ✓

FINAL CONCLUSION

AB = 2√10, M = (2, 5), line: y = 3x − 1

3x - y - 1 = 0

The calculator below opens on this line. Switch tool to try the section formula or the area of a triangle.

LABORATORY

Interactive Coordinate Geometry Calculator

LABORATORY

Coordinate Geometry Engine

What do you want to find?

Point A

Point B

Common Questions

What is the distance formula?

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. It is Pythagoras: the horizontal and vertical gaps are the two legs, and the distance is the hypotenuse.

How do you find the slope of a line through two points?

Rise over run:

m = \frac{y_2 - y_1}{x_2 - x_1}

. A vertical line (same x) has no slope; a horizontal line has slope 0.

What is the section formula?

The point dividing AB internally in the ratio m : n is

\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)

. For external division, the plus signs become minus signs. The midpoint is the case m = n = 1.

How do you tell if two lines are parallel or perpendicular?

Parallel lines have equal slopes,

m_1 = m_2

. Perpendicular lines have slopes that multiply to −1,

m_1 m_2 = -1

(for example 2 and −1/2).

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UNIT 9 • GEOMETRY & TRIGONOMETRY

Coordinate Geometry
Points, Lines & Slopes

THE CORE FORMULAS8 TOOLS

\begin{gathered} d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ M = \left(\tfrac{x_1 + x_2}{2}, \tfrac{y_1 + y_2}{2}\right) \\ m = \frac{y_2 - y_1}{x_2 - x_1} \end{gathered}

1. Label the PointsCall them

(x_1, y_1)

and

(x_2, y_2)

and keep that order all the way through.

2. Watch the SignsSubtracting a negative adds:

3 - (-2) = 5

. Brackets help.

3. Draw ItA quick sketch shows whether the answer is sensible: a positive slope rises to the right.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

For $A(1, 2)$ and $B(3, 8)$ , find the distance AB, the midpoint of AB, and the equation of line AB.

STEP 01 • DISTANCE

AB = \sqrt{(3 - 1)^2 + (8 - 2)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \approx 6.325

STEP 02 • MIDPOINT

M = \left(\frac{1 + 3}{2}, \frac{2 + 8}{2}\right) = (2, 5)

STEP 03 • SLOPE AND EQUATION

m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3

y - 2 = 3(x - 1) \;\Rightarrow\; y = 3x - 1 \;\Rightarrow\; 3x - y - 1 = 0

Check with B: $3(3) - 1 = 8$ ✓

FINAL CONCLUSION

AB = 2√10, M = (2, 5), line: y = 3x − 1

3x - y - 1 = 0

The calculator below opens on this line. Switch tool to try the section formula or the area of a triangle.

LABORATORY

Interactive Coordinate Geometry Calculator

LABORATORY

Coordinate Geometry Engine

What do you want to find?

Point A

Point B

Common Questions

What is the distance formula?

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. It is Pythagoras: the horizontal and vertical gaps are the two legs, and the distance is the hypotenuse.

How do you find the slope of a line through two points?

Rise over run:

m = \frac{y_2 - y_1}{x_2 - x_1}

. A vertical line (same x) has no slope; a horizontal line has slope 0.

What is the section formula?

The point dividing AB internally in the ratio m : n is

\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)

. For external division, the plus signs become minus signs. The midpoint is the case m = n = 1.

How do you tell if two lines are parallel or perpendicular?

Parallel lines have equal slopes,

m_1 = m_2

. Perpendicular lines have slopes that multiply to −1,

m_1 m_2 = -1

(for example 2 and −1/2).

PreviousUnit Circle

Unit 9•44 of 47

NextPercentage Calculator

Coordinate Geometry Points, Lines & Slopes

Step-by-Step Worked Example

AB = 2√10, M = (2, 5), line: y = 3x − 1

Interactive Coordinate Geometry Calculator

Coordinate Geometry Engine

Common Questions

NETZ Engine

Coordinate Geometry Points, Lines & Slopes

Step-by-Step Worked Example

AB = 2√10, M = (2, 5), line: y = 3x − 1

Interactive Coordinate Geometry Calculator

Coordinate Geometry Engine

Common Questions

Coordinate Geometry
Points, Lines & Slopes

Coordinate Geometry
Points, Lines & Slopes