Problem 49 - Intermediate Algebra

Question

Set up an equation and solve each problem. The sum of the lengths of the two legs of a right triangle is 21 inches. If the length of the hypotenuse is 15 inches, find the length of each leg.

Step-by-Step Solution

Answer

The legs are 9 inches and 12 inches long.

Step 1: Understand the Problem

We are given a right triangle with two legs whose lengths sum up to 21 inches, and a hypotenuse that is 15 inches long. We need to find the lengths of the two legs.

Step 2: Set Up Variables

Let's denote the lengths of the two legs as \(a\) and \(b\). We are given two equations: \(a + b = 21\) and the Pythagorean theorem \(a^2 + b^2 = 15^2\).

Step 3: Express One Variable in Terms of the Other

From the equation \(a + b = 21\), we can express one variable in terms of the other: \(b = 21 - a\).

Step 4: Substitute into the Pythagorean Theorem

Substitute \(b = 21 - a\) into \(a^2 + b^2 = 15^2\). The equation becomes \(a^2 + (21 - a)^2 = 225\).

Step 5: Simplify and Solve the Quadratic Equation

Expand and simplify the equation: \(a^2 + (21 - a)^2 = a^2 + (441 - 42a + a^2) = 225\). This simplifies to \(2a^2 - 42a + 441 = 225\). Subtract 225 from both sides to get \(2a^2 - 42a + 216 = 0\). Divide the entire equation by 2 to simplify further: \(a^2 - 21a + 108 = 0\).

Step 6: Factor the Quadratic Equation

The equation \(a^2 - 21a + 108 = 0\) can be factored as \((a - 9)(a - 12) = 0\).

Step 7: Solve for Possible Values of 'a'

Set each factor to zero: \(a - 9 = 0\) or \(a - 12 = 0\). Solving these gives \(a = 9\) or \(a = 12\).

Step 8: Find Corresponding Values of 'b'

If \(a = 9\), then \(b = 21 - 9 = 12\). If \(a = 12\), then \(b = 21 - 12 = 9\). Therefore, the lengths of the two legs are 9 inches and 12 inches.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that describes the relationship between the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides, often called the legs. You can express this relationship with the following formula: \[ c^2 = a^2 + b^2 \] Here, \(c\) represents the length of the hypotenuse, while \(a\) and \(b\) denote the lengths of the two legs.
In the exercise, we apply this theorem to a triangle with a hypotenuse of 15 inches. This translates to our equation as \(a^2 + b^2 = 15^2\). Knowing the sum of the legs is 21 inches allows us to set up additional equations that further help solve the problem.

Solving quadratic equations

Quadratic equations are mathematical expressions where the highest degree is squared, generally taking the form \( ax^2 + bx + c = 0 \). These equations can often be solved by various methods, including factoring, using the quadratic formula, or completing the square.
In our solution, we arrived at a quadratic equation that can be solved by factoring. Through the process of substitution and simplification using the Pythagorean theorem, we derived the quadratic form \(a^2 - 21a + 108 = 0\).

• Factoring breaks down this equation into two linear equations: \((a - 9)(a - 12) = 0\).
• Each factor individually is set to zero to find the potential values for \(a\): \(a - 9 = 0\) and \(a - 12 = 0\).
• These result in \(a = 9\) and \(a = 12\), indicating the possible lengths of one leg.

This method allows us to explore multiple possible solutions effectively.

Algebraic equations

Algebraic equations form the backbone of solving mathematical problems by establishing relationships between variables using operations such as addition, subtraction, multiplication, and division.
In the context of the exercise, we utilize algebraic equations to solve for the unknowns. The first equation, derived from the problem statement, is \(a + b = 21\). This gives us a way to express one variable in terms of the other: \(b = 21 - a\).

• Substitution into the Pythagorean theorem equation \(a^2 + (21 - a)^2 = 225\) helps to further reduce the complexity into a single variable problem.
• By methodically expanding and simplifying this expression, we establish the quadratic equation \(a^2 - 21a + 108 = 0\).

Through these sequential steps, algebra provides a structural path for finding accurate and logical solutions for the lengths of the triangle's legs.