Problem 81 - A Graphical Approach to College Algebra

Question

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$a=5, b=8 ; \text { find } c$$

Step-by-Step Solution

Answer

The length of the hypotenuse \( c \) is approximately 9.434.

Step 1: Recall the Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled 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's expressed as: \( c^2 = a^2 + b^2 \).

Step 2: Substitute Given Values

Insert the given values for \( a \) and \( b \) into the Pythagorean Theorem formula. You have \( a = 5 \) and \( b = 8 \), so the equation becomes: \( c^2 = 5^2 + 8^2 \).

Step 3: Calculate the Squares

First, calculate the square of \( a \) and \( b \). \( 5^2 = 25 \) and \( 8^2 = 64 \).

Step 4: Add the Squares

Add the results from the previous step to find the value of \( c^2 \). Thus, \( c^2 = 25 + 64 \). This simplifies to \( c^2 = 89 \).

Step 5: Solve for c

To find \( c \), take the square root of both sides of the equation. Therefore, \( c = \sqrt{89} \).

Step 6: Calculate the Value of c

Using a calculator, evaluate \( \sqrt{89} \), which is approximately \( 9.434 \). Thus, the length of the hypotenuse \( c \) is approximately 9.434.

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of its angles measures exactly 90 degrees. This right angle serves as a significant characteristic in identifying right triangles. The two sides that form this right angle are known as the "legs," and the longest side opposite the right angle is called the "hypotenuse." This hypotenuse is crucial in understanding and utilizing the Pythagorean Theorem.

In a right triangle:

• The legs are the shortest sides and meet to form the right angle.
• The hypotenuse is always the longest side, and it does not form the right angle.

The Pythagorean Theorem is only applicable to right triangles. This makes identifying them fundamental when approaching problems involving this theorem.

Hypotenuse Calculation

Calculating the hypotenuse is a straightforward application of the Pythagorean Theorem. The theorem provides a reliable method to determine the length of this longest side when the lengths of the other two sides are known. The relationship is expressed with the equation: \[ c^2 = a^2 + b^2 \]Here, \( a \) and \( b \) are the lengths of the legs, while \( c \) is the hypotenuse.

Steps to calculate include:

• Square the lengths of both legs. For instance, if \( a = 5 \) and \( b = 8 \), then \( 5^2 = 25 \) and \( 8^2 = 64 \).
• Add these squares together: \( 25 + 64 = 89 \).
• Take the square root of the result to solve for \( c \). Thus, \( c = \sqrt{89} \), which is approximately \( 9.434 \).

This calculated value is the length of the hypotenuse, offering the solution to our problem.

Geometry

Geometry involves the study of shapes, sizes, and properties of space. It is a vital mathematical discipline that provides a framework for understanding various physical spaces and forms. The Pythagorean Theorem is one of geometry's fundamental concepts, highlighting relationships within right triangles.

Key geometric principles:

• Shapes and their properties: Understanding different types of triangles, including right triangles, is essential.
• Measurements: Geometry involves calculating lengths, areas, and volumes, often using algebraic techniques.
• Spatial reasoning: Helps visualize and solve problems related to distance and space, exemplified by the hypotenuse calculations.

The application of the Pythagorean Theorem to find unknown sides of triangles is a straightforward yet profound geometric exercise, demonstrating how this theorem bridges simple algebraic manipulation with geometric reasoning.