9. The Distance Formula

Learning goals

By the end of this lesson, you should be able to:

• derive and apply the distance formula;
• distinguish exact distance, squared distance, and a rounded decimal; and
• compare squared side lengths to test a coordinate triangle for a right angle.

To find the distance between any two points, draw a right triangle: the horizontal change is one leg, the vertical change is the other, and the straight-line distance is the hypotenuse. That is just the Pythagorean Theorem on the grid:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \]

The steps:

• Find the horizontal change \((x_2 - x_1)\) and the vertical change \((y_2 - y_1)\).
• Square each, add them, take the square root.

Worked example: from \((1, 1)\) to \((5, 4)\).
\[ d = \sqrt{(5-1)^2 + (4-1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \]

Case study - the straight-line ("as the crow flies") distance between two points on a map is exactly this formula.

Exact distance versus squared distance

For points \((-3,2)\) and \((4,-2)\):
\[ \Delta x=4-(-3)=7, \qquad \Delta y=-2-2=-4. \]
Therefore,
\[ d^2=7^2+(-4)^2=49+16=65, \qquad d=\sqrt{65}\approx8.1. \]
The value 65 is the squared distance, not the actual distance. Squared distance is useful for comparing lengths without repeatedly taking square roots, but the final length is \(\sqrt{65}\).

Use squared distances to classify a triangle

Suppose three coordinate points create squared side lengths 25, 144, and 169. Because
\[ 25+144=169, \]
the triangle is right. This is the converse of the Pythagorean Theorem. Compare the two smaller squared lengths with the largest; there is no need to calculate three decimal square roots.

A safer substitution layout

Write vertical columns before simplifying:
\[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} =\sqrt{(4-(-3))^2+(-2-2)^2}. \]
Parentheses protect negative differences. Only after both differences are correct should you square and add.

Symmetry check: switching the order of the two points must give the same distance. If it does not, a sign or exponent was entered incorrectly.

Common mistake: subtracting then forgetting to square, or squaring a negative wrong. Note \((-4)^2 = 16\), not \(-16\) -- squaring always gives a positive.