Problem 46 - Elementary and Intermediate Algebra

Question

Quadratic Equation Model Problems When naming the legs of a right triangle, explain why it doesn't matter which leg you label \(a\) and which leg you label \(b\).

Step-by-Step Solution

Answer

It doesn't matter because \( a^2 + b^2 = b^2 + a^2 \) due to the commutative property.

Step 1: Recall the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle, the relationship between the lengths of the sides is given by the formula: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the lengths of the legs and \( c \) is the length of the hypotenuse.

Step 2: Analyze Impact of Switching Leg Labels

The legs \( a \) and \( b \) are interchangeable in the equation because addition is commutative. This means that \( a^2 + b^2 = b^2 + a^2 \). Thus, switching the labels of \( a \) and \( b \) does not affect the equation.

Step 3: Conclude on Labeling Flexibility

Since swapping the labels of the legs does not alter the equation, it does not matter which leg is called \( a \) and which is called \( b \). Both configurations will lead to the same \( c^2 = a^2 + b^2 \) relationship and calculation of the hypotenuse \( c \).

Key Concepts

Quadratic Equations

When we talk about quadratic equations, we're dealing with equations that involve a term with a variable squared, such as \( ax^2 + bx + c = 0 \). These equations are foundational in algebra due to their constant appearance in various mathematical problems, such as physics and engineering applications. A simple example of a quadratic equation is \( x^2 - 4 = 0 \). Solving quadratic equations often involves factoring, using the quadratic formula, or completing the square. For instance, using the quadratic formula:

• The formula is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
• It gives solutions to any quadratic equation when \( a eq 0 \).
• This formula is particularly useful when the quadratic equation is not easily factorable.

Quadratic equations are crucial for understanding the properties of parabolas, optimizing areas, and solving rate problems.

Right Triangles

Right triangles are a special category of triangles that have one angle measuring exactly 90 degrees. The side opposite the right angle is known as the hypotenuse and is always the longest side in a right triangle. The other two sides are simply referred to as the 'legs'. The relationship between these sides is beautifully summarized by the Pythagorean Theorem. This theorem states:

• For any right triangle, \( a^2 + b^2 = c^2 \),
• where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs.

Right triangles are used in a variety of practical applications, from architecture to navigation. By using the properties of right triangles, one can find distances and angles in real-world situations, which is essential in fields ranging from construction to astronomy.

Commutative Property

The commutative property is a fundamental principle in mathematics that applies to both addition and multiplication. When a property is commutative, it means you can switch the order of the numbers involved without affecting the outcome. In more formal terms:

• For addition: \( a + b = b + a \)
• For multiplication: \( a \times b = b \times a \)

In the context of the Pythagorean Theorem, this property allows flexibility in how we label the legs of a right triangle. Whether you label the shorter side as \( a \) and the other as \( b \) or vice versa makes no difference in the context of calculating \( c \), the hypotenuse. The theorem, \( a^2 + b^2 = c^2 \), remains valid regardless of the labels due to the commutative property of addition.