Coordinate Geometry • Topic 1 of 4

Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes straight from the Pythagorean theorem: the horizontal gap and vertical gap are the legs of a right triangle whose hypotenuse is the distance. Find the change in x and the change in y, square each, add, and take the square root. Many SAT problems use coordinates that form a Pythagorean triple (like 3-4-5 or 5-12-13) so the result is a whole number. Order does not matter inside the squares, since squaring removes any sign. Take the change, not the raw coordinates, before squaring.

Distance from (0,0) to (3,4) using a 3-4-5 right triangleDistance formulaO(0, 0)(3, 4)345Distance = √(3² + 4²) = 5

✅ Solved examples

1. Find the distance between (0, 0) and (3, 4).
√(3² + 4²) = √25 = 5.
2. Find the distance between (1, 2) and (4, 6).
√(3² + 4²) = √25 = 5.
3. Find the distance between (0, 0) and (5, 12).
√(25 + 144) = √169 = 13.
4. Find the distance between (2, 1) and (2, 7).
Same x; distance = |7 − 1| = 6.

✏️ Practice — try these, take hints as needed

1. Find the distance between (0, 0) and (6, 8).
√(Δx² + Δy²).
√(36 + 64).
√100.
10.
2. Find the distance between (2, 3) and (5, 7).
Δx = 3, Δy = 4.
√(9 + 16).
—
5.
3. Find the distance between (1, 1) and (13, 6).
Δx = 12, Δy = 5.
√(144 + 25).
√169.
13.
4. Find the distance between (−2, 0) and (1, 4).
Δx = 3, Δy = 4.
√25.
—
5.
5. Find the distance between (3, 5) and (3, 12).
Same x.
|12 − 5|.
—
7.

📝 Topic test — 8 questions

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