291. Distance from the Origin · always √x² + y²

Try to break it

Drag P into each quadrant. OP always equals √(x² + y²) — the squaring kills the signs, so it doesn't matter whether x or y is negative. Try to find a P where OP disagrees with the formula; impossible. The distance is always non-negative.

How you build it

Find the distance from the origin to a point.

• Point tool: mark O at the origin (0, 0) — where the axes cross.
• Point tool: plot P at (3, 2) — 3 right and 2 up from O.
• Point tool: mark F on the x-axis directly below P, at (3, 0).
• Segment tool: join O to F — the horizontal leg, length x = 3.
• Segment tool: join F to P — the vertical leg, length y = 2. The right angle is at F.
• Segment tool: join O to P — the hypotenuse. OP = √(x² + y²) = √(3² + 2²) = √13 ≈ 3.6.

The proof, step by step

Prove that the distance of a point from the origin is √(x² + y²).

• State the coordinates of the origin O.
• State the coordinates of point P.
• Write the general distance formula between two points.
• Substitute O(0,0) and P(x,y) into the formula.
• Simplify the expression to find the distance.

Worked example

Find the distance of the point P(5, 12) from the origin O(0, 0).

Using the formula d = √x² + y², we get d = √5² + 12² = √25 + 144 = √169 = 13.

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• 13 — correct
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