Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

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.

Step-by-Step Solution

Answer

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

Step 1: Understanding the Right Triangle Property

In a right triangle with hypotenuse length \( h \), we know from the Pythagorean Theorem that the square of the hypotenuse \( h \) is equal to the sum of the squares of the other two legs of the triangle. Thus, we express this as \( h^2 = x^2 + y^2 \), where \( x \) is one leg and \( y \) is the other leg of the triangle.

Step 2: Setting Up the Pythagorean Equation

We have the equation \( h^2 = x^2 + y^2 \). Our goal is to solve for \( y \) in terms of \( x \). Rearrange the equation to solve for \( y^2 \). This gives us \( y^2 = h^2 - x^2 \).

Step 3: Solving for the Length of the Other Leg

To find the length of the other leg \( y \), take the square root of both sides of the rearranged equation. Thus, \( y = \sqrt{h^2 - x^2} \).

Step 4: Defining the Function for the Other Leg

We can define the function \( L(x) \) as the length of the other leg of the triangle. Using our solution from the previous step, we define \( L(x) = \sqrt{h^2 - x^2} \).

Key Concepts

Right Triangle

A right triangle is a type of triangle that boasts a special angle, known as the right angle. This angle measures exactly 90 degrees and is a defining characteristic of a right triangle.
Right triangles have fascinating properties that make them intriguing to study. These properties are tied around their sides and the relationships between them.

• **Legs:** These are the two sides that form the right angle. In the given problem, one leg is referred to as having a length of \(x\).
• **Hypotenuse:** This is the side opposite the right angle and is the longest side of the right triangle.

Using the properties of right triangles, we can find relationships between the sides, especially using the Pythagorean Theorem, which applies specifically to them.

Hypotenuse

The hypotenuse is a uniquely significant part of a right triangle. Being the longest side, it is denoted as \(h\) in the given exercise.
The hypotenuse connects the ends of the two legs and is key to many geometric explanations and calculations.

• The Pythagorean Theorem gives rise to a critical formula involving the hypotenuse, \(h^2 = x^2 + y^2\), where \(x\) and \(y\) are the triangle’s legs.
• In any right triangle, the hypotenuse must always be equal to or longer than the other two sides.

Working with the hypotenuse allows us to derive necessary formulas to solve for unknown lengths, aiding in uncovering the elegance in geometry.

Algebraic Expressions

Algebraic expressions are a core component in mathematical problem-solving, especially within geometry. They enable us to form equations based on given variables and constants.
In the context of our exercise, algebraic expressions play a crucial role in finding the unknown side of the triangle.

• Start by expressing the Pythagorean Theorem as an algebraic equation: \(h^2 = x^2 + y^2\).
• To solve for \(y\), manipulate this expression to isolate \(y^2\): \(y^2 = h^2 - x^2\).
• Finally, further simplify it to express \(y\) by taking the square root: \(y = \sqrt{h^2 - x^2}\).

Utilizing algebraic expressions in this way not only resolves the problem but adds a powerful tool to your mathematical arsenal.