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
• Solve ACT-style distance problems

What You Will Learn Today

After this lesson, you will:

1. Apply
2. Explain why this is the Pythagorean theorem
3. Find perimeters of polygons on the plane
4. Classify triangles using computed distances
5. Solve ACT problems with distances

How Far Apart Are These Points?

How would you measure the exact distance?

Build a Right Triangle Between Points

The distance is the hypotenuse of this right triangle.

The Distance Formula Is Pythagorean

For any two points and :

Key insight:

Example: A Classic Three-Four-Five Triple

Find the distance between and .

Step 1: Compute differences

Step 2: Square, add, and take the root

Example: Negative Coordinates Still Work

Distance between and :

Quick Check: An Irrational Distance Result

Find the distance between and .

Work it out before advancing...

Subtraction Order and the Third Dimension

Order doesn't matter for distance:

But keep a consistent order — it matters for slope.

3D extension (rare on the ACT):

From Single Distances to Applications

You can now find the distance between any two points.

What comes next:

• Perimeters of polygons on the coordinate plane
• Classifying triangles by their side lengths
• ACT problems with missing endpoints
• Identifying closest pairs of points

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
• → right triangle

From our triangle: ,

Isosceles (). Right? → no

Checking Equidistance Between Three Points

Is equidistant from and ?

Distance to :

Distance to :

→ Yes, equidistant.

Quick Check: Classify This Triangle

A triangle has vertices , , .

Compute all three side lengths and classify the triangle.

Is it right? Isosceles? Both?

ACT Problem: Find a Missing Endpoint

The distance between and is .

Since -coordinates are equal, this is a horizontal distance:

ACT Problem: Identify the Closest Pair

Which pair is closest: , , ?

AB and BC are tied — both equal .

Practice: Two ACT-Style Timed Problems

Problem 1: Find the distance between and .

Problem 2: A triangle has vertices at , , and . Find its perimeter.

Target: 60 seconds each. Go!

Solutions to the Practice Problems

Problem 1:

Problem 2:

• Side 1:
• Side 2:
• Side 3:
• Perimeter

Key Takeaways and Common Mistakes

✓ Distance formula IS the Pythagorean theorem
✓ Squaring eliminates sign issues
✓ Apply three times for perimeter

Square first, then add, then root
Don't forget the final square 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 for the ACT