Right triangle

A right triangle has a special corner.

This corner is straight like a book. Two sides meet at this corner. We call these sides legs. The third side is very long. It sits across from the corner.

We call this long side the hypotenuse. You can find these shapes in a rectangle. A rectangle split in half makes two right triangles. You can use them to build things. They help people make sure corners are straight.

A right triangle is a special shape. It has one corner that is perfectly square. This corner is called a right angle.

Two sides meet at this corner. We call these sides legs. The third side is the longest. It sits across from the right angle. We call this side the hypotenuse.

Every right triangle is half of a rectangle. If you cut a square in half, you get two right triangles. These triangles will have two sides that are the same length.

There is a famous rule for these shapes. It is called the Pythagorean theorem. This rule links the lengths of the three sides. If you know two sides, you can find the third. Builders use a simple 3-4-5 rule to make sure corners are square.

Right triangles also help us study angles. This study is called trigonometry. You can also fit a right triangle inside a circle. The longest side will be the wide part of the circle. This wide part is called the diameter.

A right triangle is a very special kind of shape. It has one corner that is perfectly square. This corner is called a right angle.

We call the two sides that meet at this corner the legs. The longest side sits across from the right angle. This side has a special name: the hypotenuse. You can think of a right triangle as half of a rectangle. If you draw a line from one corner to the opposite corner, you split the rectangle into two right triangles.

There are many ways to look at how these sides work together. One way is to look at the area of the triangle. To find the area, you multiply the two legs and then take half of that number.

Another way is to look at the triangle inside a circle. Thales' theorem tells us something amazing about this. If you draw a circle and use its diameter as the base of a triangle, the top corner will always be a right angle. This means the hypotenuse is the same as the diameter of its circle.

People have studied these triangles for a very long time. One of the most famous rules is the Pythagorean theorem. This rule says that if you make a square out of each leg, their areas added together equal the area of a square made from the hypotenuse. This idea was proven in ancient times. It is even found in Euclid's Elements, which is a very famous collection of math ideas.

Some special triangles have sides that are all whole numbers. We call these Pythagorean triples.

Math experts use these shapes to study the relationship between lengths and angles. This study is called trigonometry. You can use right triangles to find the exact values of certain angles. For example, a 30-60-90 triangle or a 45-45-90 triangle are very helpful. Builders and surveyors also use these shapes in their daily work. They often use a simple 3-4-5 rule to make sure a corner is exactly 90 degrees. This helps them build straight walls and flat floors.

Right triangles also have many hidden patterns. If you draw a line from the right angle straight to the hypotenuse, you create two smaller triangles. These new triangles are similar to the big one. This means they have the same shape even if they are different sizes. You can also find the center of the circle that fits perfectly inside the triangle. This is called the inradius. There are many more ways these shapes connect to the world around us.

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 $A = \frac{1}{2}ab$.