A is a geometric shape that consists of three sides, each of different lengths, and three angles, one of which is 90 degrees. The two lengths that form the right angle are called the legs, and the longest side that stands opposite the right angle is called the . One of the most popular mathematical formulas for the hypotenuse is called the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. In other words, if we label the length of the hypotenuse as c, and the lengths of the legs as a and b, we can express this relationship mathematically as follows:

c^2 = a^2 + b^2

This formula is significant because it allows us to the length of the hypotenuse without having to measure it directly. For any right triangle whose leg lengths we do know, we can use the Pythagorean theorem to determine the length of the hypotenuse.

To understand why this formula works, we can examine the relationship between the sides of a right triangle more closely. The legs of a right triangle are perpendicular to each other, which means that they form a right angle. This right angle divides the triangle into two smaller triangles, both of which are similar in shape to the original triangle.

Each of these smaller triangles has a hypotenuse that is the same length as one of the legs of the original triangle. We can use this fact to create a system of equations that will allow us to solve for the length of the hypotenuse. Let’s call the length of the hypotenuse c, and let’s call the lengths of the legs a and b, as we did earlier. Then, we can write:

c/a = a/c-b

c/b = b/c-a

These equations are based on the fact that the smaller triangles are similar to the original triangle, and therefore their sides are proportional. If we rearrange the equations and combine them, we get:

c^2/a^2 + c^2/b^2 = 2

This is called the Pythagorean identity, and it allows us to solve for the hypotenuse in terms of the legs. Specifically, we can rearrange the equation as follows:

c^2 = a^2 + b^2

This is the Pythagorean theorem, and it is the formula we use to find the length of the hypotenuse.

There are many practical uses for the Pythagorean theorem in geometry and beyond. For example, architects and engineers use it to determine the diagonal of a rectangular building or the height of a pole, based on the measurements of the base and the height. It is also used in trigonometry, where it provides the basis for the sine and cosine functions.

In summary, the Pythagorean theorem is a simple and elegant formula that allows us to find the length of the hypotenuse in a right triangle. It is based on the relationship between the sides of a right triangle and is widely used in fields such as geometry, architecture, and trigonometry. Whether you are a student learning about basic geometry or a professional engineer solving complex problems, understanding the Pythagorean theorem is an essential skill.

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