The Distance Formula
The distance formula calculates the straight-line distance between two points on the coordinate plane using the Pythagorean theorem: d = the square root of ((x2 - x1) squared plus (y2 - y1) squared). For points (1, 2) and (4, 6), the distance is the square root of (9 + 16) = the square root of 25 = 5. This Grade 7 math skill from Saxon Math, Course 2 connects algebra and geometry, and forms the foundation for understanding the Pythagorean theorem in three dimensions, equations of circles, and coordinate geometry proofs.
Key Concepts
Property An important formula that relates distance to rate and time is the following: $$ \text{distance} = \text{rate} \times \text{time} \quad d = rt $$.
Examples You plan to ride your bike for 6 hours at an average speed of 14 miles per hour: $$ d = \frac{14 \text{ miles}}{1 \text{ hour}} \cdot 6 \text{ hours} = 84 \text{ miles} $$ Your car gets 30 miles per gallon and has a 10 gallon tank. The total distance it can go is: $$ d = \frac{30 \text{ miles}}{1 \text{ gallon}} \cdot 10 \text{ gallons} = 300 \text{ miles} $$.
Explanation This handy formula lets you predict the future of your trip! If you know how fast you are going (rate) and for how long you will be traveling (time), just multiply them together. This tells you the total distance you will cover, making it perfect for planning epic road trips or bike rides.
Common Questions
What is the distance formula?
The distance between two points (x1, y1) and (x2, y2) is d = the square root of ((x2 - x1) squared + (y2 - y1) squared). It comes directly from the Pythagorean theorem.
How do I use the distance formula step by step?
Subtract the x-coordinates and square the result. Subtract the y-coordinates and square that result. Add the two squared differences. Take the square root of the sum.
Why does the distance formula work?
The horizontal and vertical differences between two points form the legs of a right triangle. The distance between the points is the hypotenuse, found using the Pythagorean theorem: a squared + b squared = c squared.
What is the distance between (3, 1) and (7, 4)?
Horizontal difference: 7 - 3 = 4. Vertical difference: 4 - 1 = 3. Distance = square root of (4 squared + 3 squared) = square root of (16 + 9) = square root of 25 = 5.
When do students learn the distance formula?
The distance formula is introduced in Grade 7-8 as a connection between coordinate geometry and the Pythagorean theorem. Saxon Math, Course 2 covers it in Chapter 9.
What are common mistakes with the distance formula?
Students sometimes forget to square both differences before adding, or forget to take the square root at the end. Always complete all four steps in order.
How does the distance formula connect to the Pythagorean theorem?
The distance formula IS the Pythagorean theorem applied to coordinate geometry. The x-difference is leg a, the y-difference is leg b, and the distance is the hypotenuse c.