Question

The two sides of a right triangle that are not the hypotenuse are the \(\underline{?}\).

Short answer

The two sides of a right triangle that are not the hypotenuse are the legs.

Step-by-step solution

Understand the Composition of a Right Triangle

A right triangle is composed of three sides: two legs and the hypotenuse. The hypotenuse is the longest side, and it is always opposite the right angle. The other two sides are called the legs.

Identify the Sides

The two sides of a right triangle that are not the hypotenuse are the legs of the triangle.

Key concepts

Right Triangle

A right triangle is a special kind of triangle that has one angle measuring exactly 90 degrees. This 90-degree angle is referred to as the right angle. The right triangle's primary characteristic is that it follows the Pythagorean theorem, which states that if a triangle has sides of lengths 'a', 'b', and 'c' (where 'c' is the hypotenuse), then the sum of the squares of the two shorter sides ('a' squared plus 'b' squared) is equal to the square of the hypotenuse ('c' squared). This relationship can be written mathematically as:
\[ a^2 + b^2 = c^2 \].
Understanding the properties of right triangles is essential in geometry and trigonometry and finds applications in various fields such as architecture, engineering, and physics.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always located opposite the right angle. In the context of the Pythagorean theorem, the hypotenuse is represented by 'c'. It is important to note that for any right-angled triangle, the hypotenuse will be the longest edge because it lies across from the greatest angle. When calculating the length of the hypotenuse given the lengths of the legs, one would use the formula derived from the Pythagorean theorem:
\[ c = \sqrt{a^2 + b^2} \].
Knowing how to find the hypotenuse is critical not just for geometry problems but also for practical scenarios like determining the distance across a diagonal.

Triangle Legs

In a right triangle, the triangle legs are the two sides that form the right angle. These sides are referred to as 'leg A' and 'leg B' or simply 'a' and 'b' in formulas. Unlike the hypotenuse, the legs can be of any length as long as they meet at a 90-degree angle. When you have the hypotenuse's length and need to find a missing leg, you would rearrange the Pythagorean theorem accordingly:
\[ a = \sqrt{c^2 - b^2} \] or \[ b = \sqrt{c^2 - a^2} \].
Understanding the role of the legs is crucial for solving various geometric problems and construction projects where right angles are prevalent.

Geometry

Geometry is a branch of mathematics concerned with the properties and relationships of points, lines, surfaces, and shapes. It forms the foundation of all spatial understanding and design. In geometry, figures like the right triangle play a vital role due to their unique properties and theorems applicable to them. Studying geometry equips students with the ability to think logically and solve complex spatial problems. It's a field rich with applications, from graphic design to astronomy, where understanding shapes and their properties is key. With a grasp of basic geometric principles, such as those that govern the right triangle, students can tackle a diverse array of practical and theoretical challenges.