Distance Formula on the Coordinate Plane

Coordinate Geometry — Finding Distances Between Points

In this lesson:

• Apply the distance formula to find distances
• Connect the formula to the Pythagorean theorem
• Use distances: perimeter, classification, endpoints

What You Will Learn Today

1. Compute distance with
2. Derive it from the Pythagorean theorem with a grid right triangle
3. Find a polygon's perimeter from vertex coordinates
4. Classify triangles or test equidistance by comparing distances
5. Find a missing endpoint, or the closest pair of three points

How Far Apart Are These Points?

How would you measure the exact distance?

Build a Right Triangle Between Points

Legs 3 and 4, so : the hypotenuse.

The Distance Formula Is Pythagorean

Subtract point 2 minus point 1 in both coordinates:

Point order doesn't change : .

Warning: . Legs 3, 4 give 5, not 7.

Negative Coordinates and Irrational Distances

and : legs and .

and : legs 3 and 2.

Shared Coordinate: One Leg Is Zero

• Same : vertical leg , so
• Same : ; same :
• 3D: one more leg, under the root

Finding a Triangle's Perimeter on the Grid

Triangle with vertices , , .

Perimeter

Classifying Triangles With Computed Distances

• Two equal sides: isosceles
• All three equal: equilateral
• for the two shorter sides: right

Our triangle: , .

Isosceles (). Right? , so no.

Checking Equidistance Between Three Points

Is equidistant from and ?

, so yes, equidistant.

Find a Missing Endpoint on a Line

The distance between and is .

Same , so :

Identify the Closest Pair of Points

Which pair is closest: , , ?

BC is closest. No shortcut: compute and compare.

Four Lines of Work, Every Time

Distance between and :

1. Differences: ,
2. Squares: and
3. Sum:
4. Root:

Key Takeaways and Common Mistakes

• The distance formula is the Pythagorean theorem
• Subtract in one consistent order; squaring removes signs
• Same or same : one leg is zero
• Apply it once per side for perimeter

Watch out: square first, add, then take the final root;

Coming Up Next in Coordinate Geometry

Up next: The midpoint formula

• Same two points, but now we find the middle
• Midpoint connects to equidistance and bisectors
• More coordinate geometry tools