Right Triangle
GeometryA right triangle has one angle measuring exactly 90 degrees, and its sides satisfy the Pythagorean theorem: a^2 + b^2 = c^2.
Formula
a^2 + b^2 = c^2 \text{ (Pythagorean theorem)}
Definition
A right triangle has one corner that is exactly a right angle ($90^\circ$). The two sides that form the right angle are called legs, and the longest side (opposite the right angle) is called the hypotenuse.
Example
A triangle with sides $3, 4$, and $5$ is a right triangle: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. The side of length $5$ is the hypotenuse. A ramp against a wall forms a right triangle.
Key Insight
The Pythagorean theorem ($a^2 + b^2 = c^2$) only works for right triangles. This special relationship between the sides is what makes right triangles the foundation of all of trigonometry.
Definition
A right triangle contains one $90^\circ$ angle. The two shorter sides are the legs ($a$ and $b$); the side opposite the right angle is the hypotenuse ($c$). The Pythagorean Theorem states $a^2 + b^2 = c^2$. The two acute angles are complementary (sum to $90^\circ$). Trigonometric ratios (sin, cos, tan) are defined using right triangles.
Example
A $45$-$45$-$90$ triangle with legs $1$ each has hypotenuse $\sqrt{2}$. A $30$-$60$-$90$ triangle with short leg $1$ has long leg $\sqrt{3}$ and hypotenuse $2$. $\sin(30^\circ) = 1/2$, $\cos(30^\circ) = \sqrt{3}/2$, $\tan(30^\circ) = 1/\sqrt{3}$.
Key Insight
Right triangles are the engine of trigonometry: every trig ratio (sin, cos, tan, sec, csc, cot) is defined as a ratio of sides in a right triangle. Extending to all angles requires the unit circle, but the right triangle is the intuitive foundation.
Definition
A right triangle with legs $a, b$ and hypotenuse $c$ satisfies $c^2 = a^2 + b^2$. It can be inscribed in a semicircle (Thales' theorem: any triangle inscribed in a semicircle is a right triangle). The geometric mean relationships hold: the altitude to the hypotenuse $h$ satisfies $h^2 = pq$ (where $p, q$ are the hypotenuse segments), and each leg satisfies $a^2 = pc$, $b^2 = qc$.
Example
Right triangle with legs $6, 8$, hypotenuse $10$. Altitude to hypotenuse: $h = (6\cdot8)/10 = 4.8$. Hypotenuse segments: $p = 36/10 = 3.6$, $q = 64/10 = 6.4$. Check: $pq = 3.6\cdot6.4 = 23.04 = 4.8^2$. Confirmed.
Key Insight
Thales' theorem (any angle inscribed in a semicircle is a right angle) is the converse of the inscribed angle theorem for $180^\circ$ arcs. It shows that the set of right triangles inscribed in a given diameter is the semicircle itself - connecting the algebraic Pythagorean condition to the geometric circle.