Question

A right triangle has a fixed hypotenuse of length \(h\) and one leg that has length \(x\). Find a formula for the length \(L(x)\) of the other leg.

Short answer

The formula for the other leg's length is \(L(x) = \sqrt{h^2 - x^2}\).

Step-by-step solution

Understand the Pythagorean Theorem

The Pythagorean Theorem applies to right triangles and states: In a right triangle with legs of lengths \(a\) and \(b\), and hypotenuse \(c\), the equation \(a^2 + b^2 = c^2\) holds true.

Assign Variables to the Triangle Sides

Given that the hypotenuse has a fixed length \(h\) and one leg has length \(x\), let's call the unknown length of the other leg \(L(x)\).

Substitute the Known Values into the Pythagorean Theorem

Substitute the known variables into the equation from Step 1: \(x^2 + L(x)^2 = h^2\).

Isolate \(L(x)\)

To find \(L(x)\), subtract \(x^2\) from both sides of the equation:\(L(x)^2 = h^2 - x^2\).

Take the Square Root to Solve for \(L(x)\)

Solve for \(L(x)\) by taking the square root of both sides of the equation:\(L(x) = \sqrt{h^2 - x^2}\).

Key concepts

Right Triangle

A right triangle is a special type of triangle that has one angle exactly equal to 90 degrees. This creates two other important components: the two legs and the hypotenuse. Right triangles are the cornerstone of various geometrical and trigonometric studies because they provide a simple way to relate the lengths of sides to angles.

• The 90-degree angle is called the right angle. It differentiates the right triangle from other types of triangles.
• The sides adjacent to the right angle are called the legs.
• The side opposite the right angle is the longest side, known as the hypotenuse.

The Pythagorean Theorem is a key concept used in right triangles to understand the relation between its sides. This makes right triangles essential in solving problems involving distances and measurements in both simple and complex shapes.

Hypotenuse

In a right triangle, the hypotenuse is unique because it is the longest side. It sits opposite the right angle, forming the base in many calculations. The significance of the hypotenuse in a right triangle cannot be overstated: it not only helps in solving problems involving triangles but also lays the foundation for more complex geometrical applications.
To determine the length of the hypotenuse when the lengths of the other two sides are known, the Pythagorean Theorem is used like this:

• If the legs of the triangle have lengths, say, \(a\) and \(b\), the hypotenuse \(c\) can be found using the formula \(a^2 + b^2 = c^2\).
• In any right triangle, this theorem is applicable, ensuring that the hypotenuse is accurately calculated from the lengths of the two legs.
• For a direct computation, you could rearrange the formula for the hypotenuse, \(c = \sqrt{a^2 + b^2}\).

Understanding the role of the hypotenuse is fundamental when working with right triangles.

Legs of a Triangle

The legs of a right triangle are the two shorter sides that together form the right angle. These are crucial because each leg has significant implications for the calculations performed using the Pythagorean Theorem.

• In our specific problem, one of the legs has a known length \(x\), and the other is unknown.
• Unlike the hypotenuse, the legs are not directly opposite the right angle; they create the angle together.
• To find the unknown length when one leg and the hypotenuse are known, you use the Pythagorean Theorem: \(x^2 + L(x)^2 = h^2\).

By rearranging and solving for the unknown leg, the formula \(L(x) = \sqrt{h^2 - x^2}\) is derived, succinctly tying the concept of legs to their functional calculations in equations. Understanding the legs' role in these calculations is vital for solving related geometric and trigonometric problems efficiently.