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Right triangles are the workhorses of geometry. With one angle fixed at 90°, the other two must sum to 90°. Trigonometric ratios—sine, cosine, tangent—link the angles to side lengths, while the Pythagorean theorem (c² = a² + b²) guarantees the relationship between the three sides.
When a problem supplies a single side and one angle, the usual approach is to use trigonometry or the Pythagorean theorem to fill in the missing pieces. Special right triangles simplify this process because their side ratios are constants, so knowing just one side instantly reveals the entire triangle.
A 30‑60‑90 triangle is characterized by angles of 30°, 60°, and 90°. Its sides follow the fixed ratio 1 : √3 : 2, where the shorter leg (opposite 30°) is 1, the longer leg (opposite 60°) is √3, and the hypotenuse is 2. Recognizing this ratio immediately tells you the other two angles and the proportional lengths of all sides.
The isosceles right triangle has two equal angles of 45° and a right angle. Its sides are in the ratio 1 : 1 : √2, meaning the legs are congruent and the hypotenuse is √2 times the length of each leg. If you encounter a right triangle with a 45° angle, you can instantly infer the other acute angle is also 45° and apply this ratio.
When solving these triangles, the key insight is that only the ratios matter, not the absolute measurements. For example, a triangle with legs of 1 ft and 1 ft and a hypotenuse of √2 ft is a 45‑45‑90 triangle, regardless of scale. Even a triangle with legs of √17 ft and √17 ft has the same 1 : 1 : √2 proportion; its hypotenuse is √17 × √2 = √34 ft.
