Common Core Standards: Math
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Finding the length of a line segment can be incredibly easy. If we plot the two points on graph paper and find that the points form either a horizontal or a vertical line, all we have to do is count the number of boxes between the two points. Easier than finding a needle in a haystack with a giant magnet.
When the points form a slanted line, it gets a little tricky. Counting boxes isn't really an option, measuring the distance isn't exact enough, and there's no exact formula to find the distance between the two points…is there?
Your students should already be nice and comfortable with the Pythagorean Theorem and using the equation a2 + b2 = c2. If they aren't, make sure to explain that whole deal before moving on to how we can use it to calculate the distance between two points.
"What?" they'll ask. "What do two points have to do with right triangles? What witchcraft is this?"
It's not witchcraft. Actually, it's not even that hard. Students should understand that any two points that form a diagonal line can be thought of as the hypotenuse of a right triangle. If they don't believe you, draw in the horizontal and vertical legs of the triangle and then taunt them with an I-told-you-so dance. (We're just kidding about the I-told-you-so dance. That wouldn't be nice.)
Now, when given points in a coordinate system, students can imagine right triangles and calculate the distance using the Pythagorean Theorem. They'll still need to count the horizontal and vertical boxes to get the lengths of the legs—until they learn the distance formula, which is derived from the Pythagorean Theorem.
While you don't have to teach them the distance formula just yet, introducing them to the distance formula could be really useful, not to mention a fun and challenging glimpse into the future. (Now, that would be witchcraft.)
- Pythagorean Theorem on the Coordinate Plane
- Coordinate Plane: Find the Distance between Two Points
- Word Problems (Distance)
- 45-45-90 Triangle: Find the Area Given a Side