Question

Write the equation called for in each of the following statements. Refer to Appendix \(A,\) "Summary of Facts and Formulas," if necessary.Express the hypotenuse \(c\) of a right triangle as a function of its legs, \(a\) and \(b\).

Short answer

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

Step-by-step solution

Recall the Pythagorean Theorem

Remember that for a right triangle, the Pythagorean Theorem states that 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). The theorem can be written as: \( c^2 = a^2 + b^2 \).

Solve for the hypotenuse

To express the hypotenuse as a function of the legs, solve the equation from Step 1 for \( c \) by taking the square root of both sides of the equation. The result will be: \( c = \sqrt{a^2 + b^2} \).

Key concepts

Hypotenuse Function

The hypotenuse function is an important concept when dealing with right triangles. It refers to the formula that gives us the length of the hypotenuse—the longest side of the right triangle—based on the lengths of the other two sides. This is directly derived from the Pythagorean theorem.

To find the hypotenuse, we state the formula in terms of a function: \( c(a, b) = \sqrt{a^2 + b^2} \). This expression tells us that given any two positive values for \( a \) and \( b \) (the lengths of the legs of the triangle), we can calculate the length of \( c \) (the hypotenuse) by plugging them into the function. It ensures a methodical way of finding the hypotenuse for any right-angle triangle, significantly simplifying the process compared to using the Pythagorean theorem from scratch each time.

Right Triangle

A right triangle is a cornerstone figure in geometry, characterized by one angle that is exactly 90 degrees, often referred to as a right angle. This triangle's sides are typically labeled as the hypotenuse (\( c \)), the longest side, which is opposite the right angle, and the legs (\( a \) and \( b \)), which are the two sides forming the right angle.

Understanding right triangles is crucial in various mathematical contexts, from trigonometry to complex geometry problems. The relationship between the sides is consistently represented by the Pythagorean theorem. This unique property makes right triangles an invaluable tool for a range of applications in real-world problem-solving, such as construction, navigation, and even computer graphics.

Mathematical Expressions

Working with mathematical expressions is the bedrock of solving algebraic problems. A mathematical expression is a combination of numbers, variables, and operation symbols that represent a particular quantity or idea. For example, \( a^2 + b^2 \) is an expression representing the sum of the squares of \( a \) and \( b \).

Expressions are tools we use to describe mathematical relationships concisely and precisely. They can be manipulated and transformed following algebraic rules, allowing us to solve for unknown variables or simplify complex relationships. A strong grasp of mathematical expressions lays the groundwork for tackling a wide array of mathematical challenges and is indispensable for deeper exploration into more advanced topics.

Algebraic Problem Solving

At the heart of algebraic problem solving lies the ability to manipulate algebraic expressions and equations. Solving algebraic problems often involves identifying unknowns, establishing the relationship between different quantities, and systematically applying algebraic rules to find a solution.

The process usually starts with interpreting the problem and expressing it in terms of mathematical expressions or equations. Then, by employing operations such as addition, subtraction, multiplication, division, and extraction of roots (as seen with the square root), we isolate the variable of interest to arrive at the answer.

• Start with a clear understanding of the given data and what is to be found.
• Formulate the appropriate algebraic expressions or equations.
• Use inverse operations to unravel the expressions.
• Check the solution in the context of the original problem to ensure it makes sense.

Tackling algebraic problems effectively requires practice, logical reasoning, and a systematic approach. Mastery of these skills opens doors to broader fields of mathematics and their practical applications.