Pythagoras with a third coordinate: The distance between two points in 3D extends the 2D rule by adding the z-difference: square each coordinate gap, add, square-root.
IB-style question — distance between two points
Find the distance between A(1, 2, 2) and B(4, 6, 14).
Step by step
- Coordinate gaps: 3, 4, 12.
- Square, add, root.
Final answer
AB = 13.
Order doesn't matter: Each gap is squared, so (x₂ − x₁) or (x₁ − x₂) gives the same result — the negative disappears.
Average each coordinate: The midpoint of a segment is found by averaging the x's, the y's and the z's — it sits exactly halfway between the two endpoints.
IB-style question — midpoint
Find the midpoint of A(2, −1, 4) and B(6, 3, 10).
Step by step
- Average each pair.
- Simplify.
Final answer
Midpoint (4, 1, 7).
Add then halve: Add the two coordinates and divide by 2 — don't subtract (that would give a gap, not a middle).
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Given a midpoint or distance, find the missing point: If you know the midpoint and one endpoint, set the average equal to the midpoint and solve for the missing coordinate (a missing endpoint is 2M − A).
IB-style question — find the endpoint
M(4, 0, 5) is the midpoint of A(2, −2, 1) and B. Find B.
Step by step
- Each coordinate of B is 2M − A.
- Simplify.
Final answer
B = (6, 2, 9).
Check it: Average A and your B — you should get M back.
The space diagonal: The longest distance in a box (the space diagonal) joins two opposite corners — set their coordinates and use the 3D distance formula (or √(l² + w² + h²) for a box).
IB-style question — diagonal of a box
A box has edges 2, 3 and 6 along the axes, one corner at the origin. Find the length of its space diagonal.
Step by step
- Opposite corner is (2, 3, 6); distance from the origin.
Final answer
The space diagonal is 7.
Same formula, real shape: Whenever you can read off the two corner coordinates, the 3D distance formula does the work.