Problem 16 - Prealgebra

Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. $$a=?, b=12, c=19$$

Step-by-Step Solution

Answer

The missing measure \(a\) is approximately 14.7.

Step 1: Understand the Problem

We are given a right triangle with sides labeled as \(a\), \(b\), and \(c\), where \(c\) is the hypotenuse. We need to find the length of side \(a\), given that \(b = 12\) and \(c = 19\).

Step 2: Setting Up the Pythagorean Theorem

Recall the Pythagorean Theorem, which states that for a right triangle, \(a^2 + b^2 = c^2\). Substitute the known values of \(b = 12\) and \(c = 19\) into the equation: \(a^2 + 12^2 = 19^2\).

Step 3: Calculating the Squares

Calculate the squares of \(b\) and \(c\). Here, \(12^2 = 144\) and \(19^2 = 361\). Substitute these values into the equation: \(a^2 + 144 = 361\).

Step 4: Solving for \(a^2\)

Rearrange the equation to solve for \(a^2\): \(a^2 = 361 - 144\). Calculate the subtraction: \(361 - 144 = 217\). So, \(a^2 = 217\).

Step 5: Calculating \(a\)

Take the square root of both sides to solve for \(a\): \(a = \sqrt{217}\). Calculate \(\sqrt{217}\). Using a calculator, \(\sqrt{217} \approx 14.7\). Thus, \(a\) is approximately 14.7.

Key Concepts

Understanding the Right Triangle

A right triangle is a triangle that has one angle measuring 90 degrees. The side opposite this right angle is called the hypotenuse. It is the longest side of the right triangle. The other two sides are known as the legs. These are the two sides that form the right angle when they meet.
When solving problems involving right triangles, it's essential to identify these three parts because they help in applying the Pythagorean Theorem. Right triangles are a basic concept in geometry, but they serve as a foundation for understanding more complex shapes and theorem applications.

• One right angle (90 degrees)
• Two legs forming the right angle
• Hypotenuse: opposite the right angle and longest side

Understanding these components is vital for solving geometric equations and navigating through mathematical problems efficiently. Knowing how to identify and use the right triangle in equations is crucial for success.

Defining the Hypotenuse

In the context of a right triangle, the hypotenuse is the most important side when it comes to calculations. It is the longest of the three sides and directly opposite the right angle. This means it stretches across from the 90-degree angle to connect the ends of the legs.
In many problems, finding the hypotenuse is key to other calculations, such as determining area or identifying unknown side lengths. The Pythagorean Theorem uses the hypotenuse to relate all the side lengths in a formula. The theorem, expressed as \(a^2 + b^2 = c^2\), directly involves the hypotenuse (here represented as \(c\)). Calculations involving the hypotenuse are central to trigonometry and other mathematical areas.

• Always the longest side
• Critical for applying the Pythagorean Theorem
• Serves as a reference point for other calculations

Remembering that the hypotenuse is always opposite the right angle will help avoid mistakes in geometry problems.

Solving Equations: The Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry, especially when working with right triangles. It allows us to relate the lengths of the legs to the hypotenuse. The equation \(a^2 + b^2 = c^2\) is used to find one unknown side if the other two are known.
To solve an equation using this theorem, it's necessary to substitute known values into the formula. For example, if you know \(b = 12\) and \(c = 19\), you substitute these into the equation and solve for \(a\).
Here’s how it works:

• Substitute known values into the formula.
• Solve for the square of the unknown side.
• Perform necessary arithmetic (addition, subtraction).
• Take the square root to find the side length.

Through these steps, the Pythagorean Theorem helps solve various geometrical problems. It's a versatile tool for determining unknown measurements, paving the way for deeper mathematical work.