Special right triangles (30-60-90, 45-45-90) and Pythagorean triples
Special Right Triangles
Special Right Triangles is a topic in Right Triangles & Trigonometry in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.9(B), which requires students to apply the relationships in special right triangles 30-60-90 and 45-45-90 and Pythagorean triples.
The special right triangles have fixed side ratios — \(1:1:\sqrt2\) for \(45\text{-}45\text{-}90\) and \(1:\sqrt3:2\) for \(30\text{-}60\text{-}90\) — along with Pythagorean triples.
Theory
Two special right triangles have side ratios worth memorizing:
- 45-45-90 (isosceles): legs equal, hypotenuse \(=\text{leg}\cdot\sqrt2\). Ratio \(1:1:\sqrt2\).
- 30-60-90: short leg, long leg \(=\text{short}\cdot\sqrt3\), hypotenuse \(=2\cdot\text{short}\). Ratio \(1:\sqrt3:2\).
The special ratios:
How to use special triangles and triples
- Identify the triangle type or the triple.
- Match the known side to its place in the ratio.
- Scale the whole ratio by the same factor.
- Read off the missing sides.
In a 45-45-90 triangle the hypotenuse is a leg times \(\sqrt2\).
| \(\text{hyp}\) | \(=\) | \(5\sqrt2\) |
The ratio is short : long : hyp \(=1:\sqrt3:2\).
| \(\text{hyp}\) | \(=\) | \(2\cdot 6=12\) |
| \(\text{long leg}\) | \(=\) | \(6\sqrt3\) |
Check the Pythagorean relationship.
| \(8^2+15^2\) | \(=\) | \(64+225=289\) |
| \(17^2\) | \(=\) | \(289\) |
Equal, so yes — a Pythagorean triple.
\(9,12\) is \(3\) times \(3,4\), so the hypotenuse is \(3\) times \(5\).
| \(\text{hyp}\) | \(=\) | \(3\times 5=15\) |
Common pitfalls
Frequently asked questions
What is the 45-45-90 triangle ratio?
\(1:1:\sqrt2\) — the two legs are equal and the hypotenuse is a leg times \(\sqrt2\).
What is the 30-60-90 triangle ratio?
\(1:\sqrt3:2\) — short leg, long leg (short times \(\sqrt3\)), and hypotenuse (twice the short leg).
What is a Pythagorean triple?
Three whole numbers that satisfy \(a^2+b^2=c^2\), such as \(3\text{-}4\text{-}5\) or \(5\text{-}12\text{-}13\).
Are multiples of triples also triples?
Yes. Any multiple of a Pythagorean triple is also a triple, like \(6\text{-}8\text{-}10\).