Distance on the Coordinate Plane

Lesson 1 of 1: Points to Right Triangles

In this lesson:

• Build a right triangle between two points
• Apply the theorem to find the distance

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Build a right triangle between two points
2. Find each leg from the coordinates
3. Apply the theorem to find the distance
4. Handle all four quadrants and shared-coordinate cases

How Far Apart Are These Points?

Counting squares works going across or up — but the path here is diagonal.

Trace a Path Across and Up

Go 3 across, then 4 up. That L-shape is made of grid lines you can measure.

The L-Shape Is a Right Triangle

The across and up legs meet at a right angle. The segment AB is the hypotenuse.

Apply the Theorem to Get the Distance

The triangle has legs 3 and 4:

A ruler on the diagonal confirms it: 5 units.

Read the Legs Straight From Coordinates

Horizontal leg , vertical leg . Absolute value keeps lengths positive.

Distance When Coordinates Are Negative

Points E(−2, 3) and F(4, −1):

Your Turn: From the Origin

Find the distance from (0, 0) to (3, 4).

Build the triangle, read the legs, apply the theorem. Try it first.

What If the Points Share a Coordinate?

When two points share an x or a y, one leg becomes zero.

Is there still a triangle? What is the distance then?

Across Quadrants, the Same Method

M(−5, 4) and N(3, −2) sit in different quadrants.

Legs are and , so .

Same Quadrant, All Coordinates Negative

Points P(−6, −1) and Q(−2, −4):

Shared x: The Triangle Collapses

R(3, −2) and S(3, 7) share . Distance is just .

Shared y-Coordinate: Just Subtract Directly

Points T(−8, 5) and U(4, 5) share .

Straight-Line Distance on a Map

School (2, 3) to library (8, 11): legs and , straight-line distance blocks.

Why the Shortcut Beats Walking

Walking the streets: blocks. Straight line: blocks.

The straight path is always shorter than going leg by leg.

Find a Distance on Your Own

Find the distance between (−3, 2) and (5, −4).

Sketch the triangle yourself — no grid is given. Show all work.

Three Distance Traps to Avoid

Counting diagonal squares — one diagonal step is , not 1
Adding the legs gives the walking path, not the straight distance
Skipping the triangle — always sketch and label it

Every Diagonal Is a Hypotenuse

✓ Two points define a right triangle; the segment is the hypotenuse
✓ Coordinates give the legs; the theorem gives the distance
Sketch the triangle every time — don't just plug into a formula

Next: in high school, this becomes the distance formula.