math class 678 4-in-1 skill sheets
8.G.B.8 8th Grade Geometry

Distance on Coordinate Plane

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

How to explain it

At this standard, students apply the Pythagorean Theorem to find the distance between two points on the coordinate plane by identifying and squaring the horizontal and vertical legs of a right triangle.

The anchor students hold onto: Build a right triangle: horizontal leg = change in x, vertical leg = change in y. Then a² + b² = c² gives the distance c. Take √ at the end; round with ≈ if needed.

Distance on the coordinate plane unlocks the Distance Formula in high school algebra, the equation of a circle, and vector magnitude in physics — all built on a² + b² = c².

Worked examples

Example 1 Anchor (6-8-10)
Find: A(1, 2) to B(7, 10).
Step 1Horizontal: 7 − 1 = 6. Vertical: 10 − 2 = 8.
Step 26² + 8² = c².
Step 336 + 64 = 100 = c².
Answerc = √100 = 10.
Example 2 Four quadrants
Find: (−3, 1) to (5, 7).
Step 1Horizontal: 5 − (−3) = 8. Vertical: 7 − 1 = 6.
Step 28² + 6² = c².
Step 364 + 36 = 100 = c².
Answerc = √100 = 10.
Example 3 Irrational distance
Nearest tenth: (1, 1) to (4, 3).
Step 1Horizontal: 4 − 1 = 3. Vertical: 3 − 1 = 2.
Step 23² + 2² = c².
Step 39 + 4 = 13 = c².
Answerc = √13 ≈ 3.6.

Common mistakes

What students write Adding legs before squaring: 6 + 8 = 14, not the distance.
The fix Square each leg: a² + b² = c². Then take √ for the distance.
Try this A student found the distance between A(−1, 2) and B(3, 5). Here is their work: Horizontal change: 3 − (−1) = 4 Vertical change: 5 − 2 = 3 Distance = 4 + 3 = 7 Find and explain the error. Then find the correct distance.
What students write Stopping at c² = 100 without taking √ — reporting 100 as the answer.
The fix Always take √ at the end: c = √100 = 10, not 100.

Teacher tip

Head off the two predictable errors before they happen. First: Square each leg: a² + b² = c². Then take √ for the distance. Second: Always take √ at the end: c = √100 = 10, not 100.