Problem 75 - Algebra for College Students

Question

\(a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(b\) if \(c=17\) yards and \(a=15\) yards.

Step-by-Step Solution

Answer

\(b = 8\) yards.

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). Mathematically, it is expressed as \(c^2 = a^2 + b^2\). In this problem, \(c = 17\) and \(a = 15\). We need to find \(b\).

Step 2: Substitute Known Values into the Equation

Plug the known values for \(a\) and \(c\) into the Pythagorean Theorem equation: \(17^2 = 15^2 + b^2\).

Step 3: Simplify the Equation

Calculate the squares of the known values: - \(17^2 = 289\)- \(15^2 = 225\)Replace these values into the equation, resulting in: \(289 = 225 + b^2\).

Step 4: Solve for \(b^2\)

Subtract \(225\) from both sides of the equation to isolate \(b^2\):\[289 - 225 = b^2\]\[64 = b^2\]

Step 5: Solve for \(b\)

Take the square root of both sides to solve for \(b\):\[b = \sqrt{64}\]Since we are dealing with a length, we take the positive square root: \(b = 8\).

Key Concepts

Understanding Right Triangles

A right triangle is a special type of triangle where one of its angles is exactly 90 degrees. This angle is known as the right angle, and it gives the right triangle its name. The side opposite this angle is known as the hypotenuse, which is the longest side of a right triangle. The other two sides that form the right angle are called the legs.

• Right triangles have many practical applications in geometry and trigonometry.
• Many problems involving right triangles can be solved using the Pythagorean Theorem, a fundamental principle applying to these triangles.

The Pythagorean Theorem states that the square of the hypotenuse's length equals the sum of the squares of the other two sides. When solving problems involving these triangles, ensure you identify which side is the hypotenuse and which are the legs to apply the theorem correctly.

Identifying the Hypotenuse

In every right triangle, the hypotenuse is a key feature. As mentioned, the hypotenuse is always the side opposite the right angle, making it longer than the other two sides. To visualize, imagine a triangle with a corner forming a perfect "L" shape; the diagonal line that stretches across this "L" is the hypotenuse.

• Recognizing the hypotenuse helps in setting up and using the Pythagorean Theorem properly.
• In our exercise, the hypotenuse is identified as 17 yards.

Knowing its length is crucial because it is often the side that makes calculating the other sides possible. With the given exercise, once you know the lengths of one leg and the hypotenuse, you can find the other leg by rearranging the Pythagorean equation.

Working with the Simplest Radical Form

The simplest radical form involves simplifying square roots such that they remain in their most reduced form without decimals. This is an important aspect in mathematics because it allows for precise and clean expression of roots.

• When solving equations like the Pythagorean Theorem, you may need to find square roots.
• For example, in the given problem, the square root of 64 needed to be reduced.

The square root of 64 is calculated as follows:

Break Down the Square Root:

Since 64 is a perfect square, its simplest radical form is simply 8 because 8 multiplied by itself is 64.

Importance of the Simplest Form:

Using the simplest radical form results in exact calculations and simplifications in further mathematical operations. Always ensure to find and express roots in their simplest forms in exercises to achieve the most accurate results.