How far from the origin?

The idea

The distance of a point P(x, y) from the origin O(0, 0) is OP = √(x² + y²).

The point (3, 4)

The point P(3, 4) is 3 units to the right of the origin O and 4 units up.

How far is P from O, in a straight line?

• 7 units
• 5 units
• 12 units

Find points 5 units from O

The point O is fixed at the origin. Move P. Its legs are just its x- and y-coordinates, so the working is short.

Distance from the origin

OP = √(x² + y²) is the distance of the point P(x, y) from the origin O(0, 0).

This is the distance formula with the first point at (0, 0): the legs are x − 0 and y − 0, that is, the coordinates of P. Squaring removes the sign, so it works in every quadrant: for P(−3, 4), OP = √(9 + 16) = 5.

Points like (3, 4), (−3, 4) and (5, 0) are all 5 units from O, so they lie on a circle of radius 5 around O. For a point on an axis, such as (0, −7), OP = √49 = 7: just the distance along the axis.

Notes

OP = √(x² + y²) is the distance of P(x, y) from the origin O(0, 0).

Check yourself

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

Answer: 13 units

OP = √(25 + 144) = √169 = 13 units.

Find the distance of Q(−6, 8) from the origin.

Answer: 10 units

OQ = √((−6)² + 8²) = √(36 + 64) = √100 = 10 units. The minus sign disappears when you square.

Which point is exactly 5 units from the origin?

The point (x, 12) with x > 0 is 13 units from the origin. Find x.

Answer: 5

x² + 144 = 169, so x² = 25 and x = 5 (x is positive). Check: √(25 + 144) = 13.

• (2, 4). 2² + 4² = 20, so the distance is √20, which is less than 5.
• (4, 4). 4² + 4² = 32, so the distance is √32, which is more than 5.
• (−3, 4) — correct. Yes! (−3)² + 4² = 9 + 16 = 25 and √25 = 5.