Right triangle

A right triangle, also known as an orthogonal or rectangular triangle, is a specific type of triangle defined by its angles. It must contain one right angle, which measures exactly 90 degrees or $\pi/2$ radians. This angle is formed when two sides are perpendicular to one another.

In this shape, the two sides that form the right angle are called the legs, or catheti. The third side, which sits directly opposite the right angle, is called the hypotenuse. This side is always the longest side of the triangle.

Every right triangle can be understood as exactly half of a rectangle. If you take a rectangle and draw a line along its diagonal, you create two congruent right triangles. If the original rectangle is a square, the resulting triangles are isosceles. This means they have two equal sides and two equal angles. However, if the rectangle is not a square, the triangle is scalene, meaning all three sides have different lengths.

The relationship between the sides of a right triangle is governed by the Pythagorean theorem. This theorem states that the sum of the areas of the squares built on the two legs is equal to the area of the square built on the hypotenuse. In algebraic terms, if $a$ and $b$ are the legs and $c$ is the hypotenuse, then $a^2 + b^2 = c^2$. This principle was proven in antiquity and is recorded as proposition I.47 in Euclid's Elements.

When the lengths of all three sides are integers, the triangle is called a Pythagorean triangle. The set of those three integers is known as a Pythagorean triple.

Geometry provides several ways to describe the internal structure of these triangles. If you draw an altitude from the right-angle vertex to the hypotenuse, you divide the shape into two smaller triangles. These two new triangles are similar to the original triangle and to each other. This creates the right triangle altitude theorem. This theorem shows that the altitude is the geometric mean of the two segments it creates on the hypotenuse.

Furthermore, the orthocenter, which is the point where all three altitudes meet, is located exactly at the right-angled vertex.

Thales' theorem connects right triangles to the geometry of circles. The theorem states that if a triangle's base is the diameter of a circle and its third corner lies on the circle, the triangle must be a right triangle. The right angle will always be at the apex, or the corner on the circle. Conversely, the circumcircle of any right triangle will always have the hypotenuse as its diameter. This means the center of the circle, or the circumcenter, is located at the exact midpoint of the hypotenuse.

Right triangles are also fundamental to the field of trigonometry. Trigonometry is the study of the metrical relationships between the lengths and angles of shapes. The trigonometric functions for acute angles are defined as ratios of the sides of a right triangle. These ratios, such as sine, cosine, and tangent, depend only on the angle itself rather than the size of the triangle. Special triangles, such as the 30-60-90 triangle or the 45-45-90 isosceles right triangle, allow mathematicians to calculate these exact values.

Practical applications of these mathematical rules are found in many industries. In construction and surveying, professionals often use the 3-4-5 rule to ensure a corner is perfectly square. By using sides of 3, 4, and 5 units, they can confirm a 90-degree angle exists. The area of a right triangle can also be calculated simply. If one leg is used as the base, the other leg acts as the height. The area is then one half the product of these two legs, expressed as $A = \frac{1}{2}ab$.