Problem 53 - Technical Mathematics with Calculus

Question

The hypotenuse in right triangle \(A B C\), shown in Fig. \(13-3,\) is given by the Pythagorean theorem. $$ c=\sqrt{a^{2}+b^{2}} $$ Write an expression for \(c\) when \(b=3 a\), and simplify.

Step-by-Step Solution

Answer

\(c = a\sqrt{10}\)

Step 1: Rewrite the Pythagorean Theorem with Given Information

Given that in the right triangle, the relationship between the lengths of sides is such that b is three times a (b = 3a), we substitute b with 3a in the Pythagorean theorem: \(c = \sqrt{a^2 + (3a)^2}\).

Step 2: Simplify the Expression

Expand the square of 3a and simplify the expression inside the square root: \(c = \sqrt{a^2 + 9a^2} = \sqrt{10a^2}\).

Step 3: Simplify the Square Root

Since the square root of a squared term is the absolute value of the term (and because lengths are positive), we can further simplify: \(c = \sqrt{10a^2} = a\sqrt{10}\).

Key Concepts

Right Triangle

Understanding the right triangle is crucial for solving many problems in geometry, including those that involve the Pythagorean theorem. A right triangle is a type of triangle that includes one angle measuring exactly 90 degrees, known as the right angle. The sides of a right triangle have specific names: the side opposite the right angle is the hypotenuse, and the two sides that create the right angle are called the legs.

When applying the Pythagorean theorem, it's the relationship between the lengths of these sides that we're interested in, particularly, the fact that the square of the hypotenuse is equal to the sum of the squares of the other two sides, denoted by the equation: \[ c^2 = a^2 + b^2 \] where \(c\) represents the hypotenuse, while \(a\) and \(b\) represent the lengths of the legs.

When dealing with problems that give a specific ratio between the legs, like in our exercise where \(b = 3a\), it allows us to express the hypotenuse strictly in terms of one variable, simplifying the process of finding the triangle's sides.

Simplifying Square Roots

Square roots are a fundamental part of algebra and appear frequently when solving for unknowns in geometric formulas. When simplifying square roots, the objective is to find the square root of a number or expression which often involves prime factorization or recognizing perfect squares.

For instance, the expression \( \sqrt{10a^2} \) from our exercise contains both a perfect square, \(a^2\), and a non-square number, 10. Since the square root of \(a^2\) is \(a\), the expression simplifies to \(a\sqrt{10}\). Simplifying square roots also helps with further algebraic manipulations and makes it possible to evaluate the square root numerically or compare it with other numbers.

It's important to remember that when dealing with variables under the square root, we consider the positive square root because length cannot be negative. Thus, simplifying \(\sqrt{10a^2}\) to \(a\sqrt{10}\) assumes that \(a\) represents a positive quantity, consistent with its interpretation as the length of a side of a triangle.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and sometimes parentheses that represent a particular quantity or idea. In our exercise, \(c = \sqrt{a^2 + (3a)^2}\) is an expression that represents the length of the hypotenuse in terms of \(a\), the length of one of the triangle's legs.

When we deal with mathematical expressions involving square roots or powers, it's essential to carry out operations correctly. For instance, applying the exponent to the term \(3a\) within the square root requires squaring both the numeral and the variable, resulting in \(9a^2\). Combining like terms and simplifying the expression underneath the square root leads us to an easier, more workable expression. This process is not merely about making things neater; it can also reveal relationships and dependencies between variables that are not immediately obvious.

In our geometry problem, simplifying the expression for the hypotenuse has transformed it from a radical expression containing a binomial to a radical expression involving a single term. Through these simplifications, we get closer to representing real-world quantities, like the length of the hypotenuse, in a clear and concise manner.