Problem 31 - Algebra 2 and Trigonometry

Question

The length of the longer leg, \(a,\) of a right triangle is 1 centimeter less than the length of the hypotenuse and the length of the shorter leg, \(b,\) is 8 centimeters less than the length of the hypotenuse. a. Express \(a\) and \(b\) in terms of \(c,\) the length of the hypotenuse. b. Express \(a^{2}+b^{2}\) as a polynomial in terms of \(c\) . c. Use the Pythagorean Theorem to write a polynomial equal to \(c^{2}\) .

Step-by-Step Solution

Answer

a) \(a = c - 1, b = c - 8\). b) \(a^2 + b^2 = 2c^2 - 18c + 65\). c) \(c^2 = 2c^2 - 18c + 65\).

Step 1: Identify Variables

We are given that the longer leg, \(a\), is 1 cm less than the hypotenuse, \(c\), and the shorter leg, \(b\), is 8 cm less than the hypotenuse. Thus, we can express these as: \(a = c - 1\) and \(b = c - 8\).

Step 2: Expand \(a^2 + b^2\)

Substitute the expressions for \(a\) and \(b\) into the equation \(a^2 + b^2\). \(a^2 = (c-1)^2 = c^2 - 2c + 1\) \(b^2 = (c-8)^2 = c^2 - 16c + 64\) Therefore, \(a^2 + b^2 = (c^2 - 2c + 1) + (c^2 - 16c + 64)\).

Step 3: Simplify \(a^2 + b^2\)

Combine like terms to simplify the expression for \(a^2 + b^2\): \(a^2 + b^2 = c^2 - 2c + 1 + c^2 - 16c + 64\) = \(2c^2 - 18c + 65\).

Step 4: Apply the Pythagorean Theorem

According to the Pythagorean Theorem, in a right triangle \(c^2 = a^2 + b^2\). From the expression we derived for \(a^2 + b^2\), we set: \(c^2 = 2c^2 - 18c + 65\). Subtract \(c^2\) from both sides to form a polynomial equal to zero: 0 = \(c^2 - 18c + 65\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of terms combined using addition, subtraction, and multiplication, but no division by a variable. Each term is composed of a coefficient, variable, and non-negative integer exponent. For instance, in the exercise given, we derived the polynomial expression \(a^2 + b^2 = 2c^2 - 18c + 65\). Here,

• \(c^2\) and \(-18c\) are terms with degrees of 2 and 1, respectively.
• The constant term is 65.

Polynomials are useful because they allow us to model and solve a wide range of mathematical problems. By expressing problems in polynomial form, like \(c^2 - 18c + 65\), we create equations that can be solved to find variable values. Each operation retains the polynomial structure, meaning we can easily perform further calculations or manipulations.

Algebraic Expressions

Algebraic expressions, such as \(c - 1\) and \(c - 8\), are combinations of numbers, variables, and arithmetic operations. These expressions can model relationships between different quantities and allow us to express one variable in terms of another.

• When we say \(a = c - 1\), we define the relationship between the longer leg \(a\) and the hypotenuse \(c\).
• Likewise, \(b = c - 8\) reveals how the shorter leg \(b\) relates to \(c\).

Algebraic expressions are fundamental in expressing any mathematical problem clearly and succinctly. They serve as a backbone for forming equations and helping us understand how changing one variable affects another. Simplifying expressions like turning \(a^2 + b^2\) into \(2c^2 - 18c + 65\) helps to identify and solve equations effectively.

Right Triangle Geometry

Right triangle geometry focuses on triangles where one angle measures 90 degrees. The two legs and the hypotenuse form the three sides of the triangle. The Pythagorean Theorem is central in right triangle geometry. It states that the square of the hypotenuse \(c\) equals the sum of the squares of the other two sides (legs), denoted as \(a^2 + b^2\).

• Using this theorem helps us find missing side lengths given two sides.
• In this problem, the theorem allows us to derive the equation \(c^2 = a^2 + b^2\).

When we consider \(a = c - 1\) and \(b = c - 8\), substituting these into \(a^2 + b^2\) lets us utilize the Pythagorean Theorem. It reaffirms the relationship: any right-angled triangle will conform to this rule, facilitating calculations. By using the original exercise expressions, and combining them algebraically, we find \(a^2 + b^2 = 2c^2 - 18c + 65\), enabling us to create and solve polynomial equations.