Problem 15 - Elementary and Intermediate Algebra

Question

The lengths of two sides of the right triangle \(A B C\) are given. Find the length of the missing side. \(a=6 \mathrm{ft}\) and \(b=8 \mathrm{ft}\) (RIGHT TRIANGLE CAN'T COPY)

Step-by-Step Solution

Answer

The length of the hypotenuse is 10 ft.

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)). The formula is: \(c^2 = a^2 + b^2\).

Step 2: Plug in Known Values

Substitute the given side lengths \(a = 6\) ft and \(b = 8\) ft into the Pythagorean Theorem.\[c^2 = 6^2 + 8^2\]

Step 3: Calculate Squares of Known Sides

Calculate \(6^2\) and \(8^2\).\[6^2 = 36\quad 8^2 = 64\]

Step 4: Sum the Squares

Add the results from Step 3.\[36 + 64 = 100\]

Step 5: Solve for the Hypotenuse

Since \(c^2 = 100\), find \(c\) by taking the square root.\[c = \sqrt{100} = 10\]

Step 6: Conclude the Result

The length of the hypotenuse \(c\) is \(10\) ft.

Key Concepts

Understanding the Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This 90-degree angle is known as a right angle. In a right triangle, the side opposite the right angle is called the hypotenuse and is the longest side of the triangle. The two other sides are known as the legs of the triangle.

• Right triangles are unique because they follow the Pythagorean Theorem, which is a fundamental principle in geometry.
• The theorem helps in finding the length of one side, if the lengths of the other two sides are known.

Recognizing a right triangle is crucial since it allows us to apply specific mathematical principles effectively. In this exercise, triangle ABC is a right triangle with sides given, enabling us to use these methods to find unknown side lengths.

Determining the Hypotenuse

The hypotenuse is the longest side of a right triangle, found directly opposite the right angle. Identifying this side is important because it helps to apply the Pythagorean Theorem correctly.

• For the triangle ABC, with given legs of 6 feet and 8 feet, we aim to find the length of the hypotenuse.
• By using the Pythagorean Theorem: \[c^2 = a^2 + b^2\]
• The hypotenuse \(c\) can be calculated, leading to the final solution.

Once the squares of the legs are added, the square root of the sum gives the length of the hypotenuse. This exercise highlights why understanding the hypotenuse is key to trigonometry and geometry.

Algebraic Problem-Solving in Geometry

Solving problems like finding the missing side of a right triangle often involves algebraic methods. Algebraic problem-solving is the backbone of modern mathematics, enabling us to manipulate formulas to discover unknown values.

• In practice, you substitute known values into established formulas like the Pythagorean Theorem.
• This process involves squaring numbers, adding them, and then finding square roots, as seen in the steps of our exercise.
• This step-by-step approach ensures solutions are logical and verified.

By being methodical, algebraic problem-solving becomes a powerful tool, not just in geometry but across various math disciplines. It lays the groundwork for solving complex equations and enhances critical thinking skills in students.