Problem 89 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ a^{2}+b^{2}=c^{2} ; \quad \text { for } b $$

Step-by-Step Solution

Answer

\( b = \sqrt{c^2 - a^2} \)

Step 1: Understand the Equation

The equation provided is the Pythagorean theorem: \( a^2 + b^2 = c^2 \), which describes the relationship in a right triangle.

Step 2: Isolate \( b^2 \)

To solve for \( b \), first isolate \( b^2 \) on one side of the equation: \( a^2 + b^2 = c^2 \). Subtract \( a^2 \) from both sides to get \( b^2 = c^2 - a^2 \).

Step 3: Solve for \( b \)

Take the square root of both sides to solve for \( b \): \( b = \sqrt{c^2 - a^2} \). For a real solution, ensure that \( c^2 \geq a^2 \).

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)). This relationship is expressed by the equation:

• \( a^2 + b^2 = c^2 \)

This theorem is essential for solving problems involving right triangles and forms the basis of trigonometry.
When using this theorem, make sure you identify which side is the hypotenuse, as it is always opposite the right angle. This is crucial for applying the theorem correctly.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to solve for a specific variable. In this exercise, you need to isolate \( b \) in the Pythagorean equation \( a^2 + b^2 = c^2 \). To do this, you perform algebraic operations:

• First, subtract \( a^2 \) from both sides to get \( b^2 = c^2 - a^2 \).
• Then, take the square root of both sides to solve for \( b \), giving you \( b = \sqrt{c^2 - a^2} \).

This step-by-step rearrangement is fundamental to analytical skills in mathematics.
Ensure that you perform each operation on both sides of the equation to maintain equality. This concept is widely used for solving various types of equations in algebra and calculus.

Right Triangle

A right triangle is a type of triangle that has one angle measuring 90 degrees. The side opposite this right angle is known as the hypotenuse. The other two sides are often referred to as the base and height of the triangle.
In problems involving right triangles, identifying these sides correctly is pivotal because they determine how you apply formulas like the Pythagorean theorem.

• The hypotenuse is always the longest side.
• Look for the right angle to find it and apply the correct formulas.

Right triangles have unique properties that make them very useful in various fields, including engineering, architecture, and astronomy.
Understanding the structure and properties of right triangles helps simplify complex problems and is key to mastering geometric principles.