Coordinate Geometry • Topic 4 of 5

Distance & Midpoint

The distance between two points is the Pythagorean theorem in disguise: d = √[(x₂ − x₁)² + (y₂ − y₁)²], where the coordinate differences are the two legs of a right triangle and the distance is the hypotenuse. Because it is Pythagoras, the GRE's favourite point pairs give clean answers through 3-4-5 and 5-12-13 triangles — recognising those saves the calculator. The midpoint is simply the average of the coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2), the point exactly halfway between the two. A common reverse question gives one endpoint and the midpoint and asks for the other endpoint — solve each coordinate equation (the midpoint's x is the average of the two x's). Squaring makes signs disappear inside the distance formula, so (x₂ − x₁)² is the same whichever way you subtract; that forgiveness does not extend to the midpoint, where you must add, not subtract.

Two points joined by a segment forming a 3-4-5 right triangle for the distance formulaDistance = Pythagorasxy-1123456-11234567O(1, 2)(4, 6)34d = 5d = √(3² + 4²) = √25 = 5

✅ Solved examples

1. Find the distance between (1, 2) and (4, 6).
d = √[(4 − 1)² + (6 − 2)²] = √[9 + 16] = √25 = 5. (A 3-4-5 triangle.)
2. Find the midpoint of the segment joining (−2, 3) and (6, 7).
M = ((−2 + 6)/2, (3 + 7)/2) = (4/2, 10/2) = (2, 5).
3. Find the distance between (−1, −1) and (2, 3).
d = √[(2 − (−1))² + (3 − (−1))²] = √[3² + 4²] = √25 = 5.
4. M = (5, 4) is the midpoint of AB, and A = (2, 1). Find B.
(2 + x)/2 = 5 → x = 8; (1 + y)/2 = 4 → y = 7. So B = (8, 7).

✏️ Practice — try these, take hints as needed

1. Find the distance between (0, 0) and (6, 8).
√(x² + y²).
√(36 + 64).
√100.
10
2. Find the midpoint of (3, −4) and (9, 2).
Average each coordinate.
((3 + 9)/2, (−4 + 2)/2).
(6, −1).
(6, −1)
3. Find the distance between (2, 5) and (5, 9).
Differences 3 and 4.
√(9 + 16).
√25.
5
4. M = (0, 0) is the midpoint of PQ and P = (−3, 4). Find Q.
Midpoint is the average.
(−3 + x)/2 = 0.
Solve each coordinate.
Q = (3, −4)
5. Find the distance between (1, 1) and (13, 6).
Differences 12 and 5.
√(144 + 25).
√169.
13

📝 Topic test — 8 questions

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