Find Distance Between Two Points on a Coordinate Plane

Students apply the Pythagorean Theorem to find the distance between any two points (x₁, y₁) and (x₂, y₂) on a coordinate plane by treating the horizontal and vertical differences as the legs of a right triangle. They derive and use the distance formula and recognize it as an application of the theorem.

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Grade 8 · Geometry: Pythagorean Theorem · question 1 of 3

0 of 20

20 rungs. A star at the top.

(2, 2) and (5, 6): distance =

How it is taught

0:00 of 3:35

The straight distance is the hypotenuse of a right triangle.

The legs are the horizontal and vertical gaps.

Read it: Distance between two points

From P(1, 1) to Q(16, 21): walk 15 across to the corner C(16, 1), then 20 up to Q. Those walks are the legs of a right triangle, and the straight line PQ is its longest side: 15 × 15 + 20 × 20 = 625, and 25 × 25 = 625, so PQ = 25.

Walk from one point straight across to the corner level with it, then straight up or down to the other point.
Count the walk across and the walk up: those are the legs.
Square the legs, add, and find the number whose square that is.

Across is level and up is straight up, so they meet at a square corner.

The straight line between the points is across from that corner: the longest side.

225 + 400 = 625, and 25 × 25 = 625, so the points are 25 apart.

The distance between two points: the walk across and the walk up are legs; add their squares and find the number whose square it is.

Adding the walks: 15 + 20 = 35.

The straight line is shorter than the walk around the corner: 225 + 400 = 625, and 25 × 25 = 625, so PQ = 25.

(3, 2) and (23, 23): distance = ___ — 29. Across 20 and up 21 are the legs: 400 + 441 = 841, and 29 × 29 = 841.

Distance from (1, 2) to (4, 6)

How do we find the distance between two points on a coordinate plane? Let's use the Pythagorean Theorem!

First, we set up our coordinate plane with an x-axis and a y-axis.

Grade 8 · Pythagorean Theorem on the coordinate plane

Plot our first point, A, at one comma two.

Plot our second point, B, at four comma six.

Draw a horizontal line from A. Its length is four minus one — that's three.

Draw a vertical line up to B. Its length is six minus two — that's four.

Look — we made a right triangle! The distance from A to B is the hypotenuse.

Time for the Pythagorean Theorem: a squared plus b squared equals c squared.

Plug in our legs: three squared plus four squared equals c squared.

Three squared is nine, four squared is sixteen. Together, that's twenty-five.

So c is the square root of twenty-five, which is five.

The distance from A to B is five units. That's the distance formula in action!

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MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

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MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

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