Apply distance formula: √(x₂-x₁)² + (y₂-y₁)². Here the answer is 5.

Find the Distance Between (3, 4) and (0, 0) — Distance Formula

Question

Find the distance between the points (3, 4) and (0, 0).

Solution — Step by Step

For two points $(x_1, y_1)$ and $(x_2, y_2)$ , the distance between them is:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

This comes directly from the Pythagorean theorem — the distance is the hypotenuse of a right triangle formed by the horizontal and vertical gaps.

Here, $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (3, 4)$ .

We’re measuring from the origin, which simplifies things — but use the full formula anyway to build the habit.

d = \sqrt{(3 - 0)^2 + (4 - 0)^2}

d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}

d = \sqrt{25} = \mathbf{5}

The distance between (3, 4) and the origin is 5 units.

Why This Works

The distance formula is just Pythagoras in disguise. If we plot (3, 4) and (0, 0) on a grid, we can draw a right triangle: one leg runs 3 units along the x-axis, the other runs 4 units up the y-axis. The distance between the two points is the hypotenuse.

By Pythagoras: $h = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$ . This is the classic 3-4-5 right triangle, one of the most important Pythagorean triplets in board exams.

Recognising (3, 4, 5) as a triplet saves time in MCQs — you don’t even need to compute $\sqrt{25}$ if you spot it early.

Pythagorean Triplets to memorise: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). When you see two points whose differences match a triplet, the distance is the third number — no calculator needed.

Alternative Method — Using the Origin Shortcut

When one point is the origin (0, 0), the formula collapses to:

d = \sqrt{x^2 + y^2}

For (3, 4): $d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$ .

This is worth knowing for speed, but only works when one endpoint is the origin. In every other case, we must subtract the coordinates first.

Common Mistake

Forgetting to subtract before squaring. Many students write $\sqrt{x_2^2 + y_2^2 - x_1^2 - y_1^2}$ instead of $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . These are NOT the same. For example, with points (5, 0) and (3, 0), the correct distance is $\sqrt{(5-3)^2} = 2$ , but the wrong formula gives $\sqrt{25 - 9} = 4$ . Always subtract first, then square.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using the Origin Shortcut

Common Mistake

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