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SAT Topic 3

Pythagoras & distance

a² + b² = c² and the distance formula on the coordinate plane.

Concept

The Pythagorean theorem: in any right triangle, a² + b² = c² where c is the hypotenuse (the side opposite the right angle).

The distance formula is just Pythagoras applied to coordinates:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Treat Δx = x₂ − x₁ and Δy = y₂ − y₁ as the two legs of a right triangle whose hypotenuse is the distance.

Pythagorean triples to memorise — they make life much easier on the test:

Worked example 1

A right triangle has legs of length 9 and 12. Find the hypotenuse.

Solution
Pythagoras. c² = 9² + 12² = 81 + 144 = 225
Square root. c = √225 = 15
Hypotenuse 15. (You can also recognise that 9-12-15 is the 3-4-5 triple scaled by 3.)

Worked example 2

Find the distance between the points (−1, 2) and (4, 14).

Solution
Differences. Δx = 4 − (−1) = 5, Δy = 14 − 2 = 12.
Pythagoras. d = √(5² + 12²) = √(25 + 144) = √169
d = 13. (5-12-13 strikes again.)

Practice test

8 questions on the Pythagorean theorem, distance formula, and recognising classic triples.

Practice test · 8 questions Question 1 of 8 · Score 0