Right Triangle

Precise definition

A right triangle has one $90^\circ$ angle. The side opposite it is the hypotenuse and is longest; the other sides are legs. For leg lengths $a,b$ and hypotenuse $c$, the Pythagorean theorem gives $a^2+b^2=c^2$, and its converse identifies a right triangle from side lengths.

Notation and mathematical language

For acute angle $\theta$, $\sin\theta=\text{opposite}/\text{hypotenuse}$, $\cos\theta=\text{adjacent}/\text{hypotenuse}$, and $\tan\theta=\text{opposite}/\text{adjacent}$. Labels opposite and adjacent depend on the chosen angle; hypotenuse does not.

Conceptual picture

Similarity explains why trigonometric ratios depend only on angle, not triangle size. Pythagoras encodes Euclidean distance and connects to the unit circle. The two acute angles are complementary.

Fully worked example

Interpretation and application

Right triangles model surveying, construction, vectors, navigation, and coordinate distance. The perpendicular and straight-line assumptions must be validated; real slopes and surfaces may require measurement uncertainty.

Common mistakes