Right triangle

A right triangle is a special kind of triangle where two sides meet at a right angle, which is exactly 90 degrees. This right angle makes the triangle very useful in many areas of math and everyday life. The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are called legs.

Right triangles are often thought of as half of a rectangle split along its diagonal. When the rectangle is a perfect square, the right triangle formed is special because it has two sides of equal length. These triangles are very important in geometry and help us understand relationships between angles and sides.

One of the most famous rules in math, the Pythagorean theorem, applies to right triangles. It tells us that if we know the lengths of the two legs, we can find the length of the hypotenuse by using the formula a² + b² = c². When the sides of a right triangle are whole numbers and follow this rule, they are called a Pythagorean triple. Right triangles also form the basis of trigonometry, which helps us understand how angles and side lengths are related in many real-world situations.

Principal properties

Main article: Pythagorean theorem

A right triangle has one angle that is exactly 90 degrees. The side opposite this right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are called legs.

The sides of a right triangle are related by the Pythagorean theorem. This theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This important relationship helps us understand how the sides of a right triangle work together.

Characterizations

A right triangle is a special kind of triangle where two sides meet at a right angle, which is a 90-degree angle. The side opposite this right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are called legs.

Right triangles have some interesting properties. For example, the triangle can fit perfectly inside a semicircle, with the hypotenuse matching the diameter of the semicircle. Also, the center of the circle that passes through all three vertices of the triangle (called the circumcenter) is located exactly at the midpoint of the hypotenuse.

Trigonometric ratios

Main article: Trigonometric functions – Right-angled triangle definitions

The trigonometric functions help us understand angles in a right triangle. By looking at the lengths of the sides—opposite, adjacent, and hypotenuse—we can find special ratios for any angle. These ratios stay the same no matter the size of the triangle, because all such triangles are similar in shape.

For example, if we pick an angle α, we can use the side lengths to find the sine, cosine, and tangent of that angle. These ratios tell us how the sides relate to each other and to the angle we are studying.

Special right triangles

Main article: Special right triangles

Some right triangles have special angles that make calculations easier. For example, a 30-60-90 triangle helps us understand angles that are multiples of 30 degrees, and a 45-45-90 triangle (which is also an isosceles right triangle) helps with angles that are multiples of 45 degrees.

There is also a special triangle called the Kepler triangle. This triangle has sides that are related to the golden ratio, a number that appears in many areas of art and nature. The sides of the Kepler triangle follow a specific pattern called a geometric progression, making it unique among right triangles.

Thales' theorem

Main article: Thales' theorem

Thales' theorem tells us that if you have a circle and draw a line through the widest part of the circle (called the diameter), any point you pick on the circle that isn’t on that line will always form a right triangle with the two ends of the diameter. This means the angle at that point will always be a right angle, or 90 degrees.

An interesting fact that goes along with this is that the longest side of a right triangle (called the hypotenuse) will always match the diameter of the circle that passes through all three points of the triangle.

Medians

In a right triangle, special rules apply to the lines called medians, which connect each vertex to the middle of the opposite side. One important rule is that the square of the median to one leg plus the square of the median to the other leg equals five times the square of the median to the hypotenuse. This can be written as ( m_a^2 + m_b^2 = 5m_c^2 ).

Another interesting fact is that the median drawn to the hypotenuse divides the triangle into two smaller triangles that are both isosceles. This happens because this median is exactly half the length of the hypotenuse.

Euler line

In a right triangle, the Euler line passes through two important points. It goes through the vertex where the right angle is and also through the midpoint of the side opposite that vertex, which is called the hypotenuse. This happens because the point where the triangle's altitudes meet is at the right-angled vertex, and the point where the perpendicular bisectors of the sides meet is at the midpoint of the hypotenuse.

Main article: Euler line
Further information: Perpendicular bisectors of sides

Inequalities

In a right triangle, there are special rules about the sizes of its sides. The diameter of the circle that fits inside the triangle (called the incircle) is always smaller than half the longest side, which is called the hypotenuse.

There are also rules connecting the two shorter sides (called legs) with the hypotenuse and the height dropped to the hypotenuse. These rules help us understand how the sides relate to each other in different right triangles.

Other properties

A right triangle has special features. It is the only triangle that can have two different inscribed squares instead of just one or three.

There are also special rules connecting the sizes of the triangle and these squares, as well as the circle that fits perfectly inside the triangle. For example, if you know the lengths of the sides of the two inscribed squares and the length of the triangle's longest side (the hypotenuse), there are formulas that relate them all.