Problem 28 - Prealgebra

Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. $$a=8.1, b=3.5, c=? \quad$$

Step-by-Step Solution

Answer

The hypotenuse measures approximately 8.8 units.

Step 1: Understand the problem

We are given a right triangle with legs measuring \(a = 8.1\) and \(b = 3.5\), and we need to find the measure of the hypotenuse, denoted by \(c\). We will use the Pythagorean theorem to find \(c\).

Step 2: Set up the Pythagorean theorem equation

The Pythagorean theorem states that \(a^2 + b^2 = c^2\) for a right triangle. We substitute the given values into this equation: \((8.1)^2 + (3.5)^2 = c^2\).

Step 3: Calculate \(a^2 + b^2\)

Calculate each square: \(8.1^2 = 65.61\) and \(3.5^2 = 12.25\). Then add them: \(65.61 + 12.25 = 77.86\).

Step 4: Solve for \(c\)

Take the square root of both sides to solve for \(c\): \(c = \sqrt{77.86}\).

Step 5: Compute the square root

Use a calculator to find the square root: \(c \approx \sqrt{77.86} \approx 8.8\).

Key Concepts

Understanding the Right Triangle

A right triangle is a special type of triangle that has one angle measuring 90 degrees. This angle is called the right angle, and it is a crucial element in distinguishing right triangles from other types of triangles. In a right triangle, the two sides that form the right angle are known as the legs. These are usually symbolized by the letters \(a\) and \(b\). The side opposite the right angle is known as the hypotenuse, symbolized by \(c\).
The hypotenuse is always the longest side of the right triangle because it stretches across from one leg to the other opposite the right angle. Understanding which side is the hypotenuse in a right triangle is key to applying the Pythagorean theorem effectively.

Exploring the Hypotenuse

The hypotenuse is the side of a right triangle opposite the right angle, and it is also the longest side. In a geometric sense, it is the stretch of the triangle. Calculating the hypotenuse is often the focus of many mathematical problems involving right triangles because it is central to the triangle's structure.
In practical applications, understanding the hypotenuse is essential because it helps us determine distances and measurements in various fields, such as engineering and architecture. Using the Pythagorean theorem, given the lengths of the two legs of a right triangle, we can calculate the hypotenuse by solving the equation \(c = \sqrt{a^2 + b^2}\). This formula is powerful in helping us find distances across various contexts.

The Process of Squaring Numbers

Squaring a number means multiplying that number by itself. In mathematical terms, the square of \(a\) is expressed as \(a^2 = a \times a\). Squaring numbers is a basic yet vital operation in mathematics, playing an important role in the Pythagorean theorem.

• Squaring a positive number results in a positive value. For example, \(8.1^2 = 8.1 \times 8.1 = 65.61\).
• Squaring is used in the Pythagorean theorem to handle the sides of the right triangle as it relates the squares of the two legs to the square of the hypotenuse.

When solving problems involving right triangles, ensuring accurate squaring of the side lengths is crucial for achieving the correct results. Once the squares are calculated, the next step often involves summing them up or finding a square root, as demonstrated in finding the hypotenuse.