Right Triangle

Dependencies: Triangle, Perpendicular. A derived definition: add to a triangle the condition that one interior angle is right.

Definition

Let $\triangle ABC$ be a triangle.

If one interior angle is a right angle, equivalently if two sides are perpendicular, then the triangle is a right triangle. For example, $CA\perp CB$ and $\angle C=90^\circ$. The sides $CA$ and $CB$ forming the right angle are the legs, and the opposite side $AB$ is the hypotenuse.

Right triangles are the starting point of trigonometry. Fixing one $90^\circ$ angle creates calculable relationships among sides and angles, including the Pythagorean theorem and trigonometric ratios.

Remarks

• The hypotenuse is the unique longest side. The right angle is the largest angle, so its opposite side is longest. A nondegenerate triangle cannot contain two right angles.
• Pythagorean theorem. The sum of the squares of the legs equals the square of the hypotenuse (Pythagorean theorem).
• The acute angles are complementary. After subtracting $90^\circ$ from the $180^\circ$ angle sum, the remaining two angles total $90^\circ$.
• Median to the hypotenuse. Its length is half the hypotenuse (Right triangle: median to hypotenuse = half hypotenuse), equivalently the right-angle vertex lies on the circle with the hypotenuse as diameter.