The Pythagorean Theorem
Grades 8–10 Geometry Triangles
In a right triangle, the square on the longest side — the hypotenuse — equals the sum of the squares on the two legs:
a² + b² = c²
Using it both ways
Legs 3 and 4, find hypotenuse:
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
Hypotenuse 13, one leg 5, find the other:
b² = 13² − 5² = 169 − 25 = 144
b = √144 = 12
Triples worth memorizing
(3, 4, 5) · (5, 12, 13) · (8, 15, 17) · (7, 24, 25) — and any whole multiple: (6, 8, 10) is just (3, 4, 5) ×2.
The distance formula is this in disguise
Between (x₁, y₁) and (x₂, y₂): the horizontal gap and vertical
gap are the legs — d = √((x₂−x₁)² + (y₂−y₁)²). From (1, 2) to
(4, 6): √(9 + 16) = 5.
Watch out
c is always the hypotenuse — the side across from the
right angle. If you're solving for a leg, subtract the squares,
don't add them.
Practice
Legs 6 and 8 → hypotenuse = ___
Hypotenuse 13, leg 5 → other leg = ___
Is 5, 12, 13 a Pythagorean triple?
Distance from (0, 0) to (3, 4) = ___
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