Problem 48 - Algebra 1

Question

Write an equation or an inequality to model the real-life situation. The square of the length \(c\) of the hypotenuse of a right triangle is equal to four squared plus three squared.

Step-by-Step Solution

Answer

The equation modeling the real-life situation is \(c^2 = 4^2 + 3^2\), and the length of the hypotenuse is \(c = 5\).

Step 1: Understanding the problem

The problem is a classic example of a right triangle situation where the Pythagorean theorem applies. In this situation, the two sides are given as 4 and 3, and the hypotenuse is \(c\).

Step 2: Application of the Pythagorean theorem

According to the Pythagorean theorem, the square of the hypotenuse length \(c^2\) is equal to the sum of the squares of the lengths of the other two sides. In this case, these sides have lengths 4 and 3. Therefore, the equation is \(c^2 = 4^2 + 3^2\).

Step 3: Solving the equation

By solving the equation \(c^2 = 4^2 + 3^2\) we find that \(c^2 = 16 + 9 = 25\), and by taking the square root of both sides, we get \(c = 5\). So, the length of the hypotenuse is 5 units.

Key Concepts

Understanding Right Triangles

Right triangles are a special class of triangles that have one angle exactly equal to 90 degrees, called the right angle. The sides that form the right angle are known as the 'legs' of the triangle, while the side opposite the right angle is called the 'hypotenuse.' The hypotenuse is always the longest side of a right triangle.

The properties of right triangles are often studied in geometry due to their importance in various fields, including architecture, engineering, and physics. They also serve as the foundation for trigonometry, which is the study of the relationships between the angles and sides of triangles. Understanding right triangles is essential for solving problems that involve right-angled structures or components.

Calculating Hypotenuse Length

In the context of right triangles, the hypotenuse length is of particular interest because it connects the two legs at the right angle. To find the hypotenuse length, we utilize the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (\(c^2\)) is equal to the sum of the squares of the other two sides (\(a^2 + b^2\)).

When given the lengths of the legs, calculating the hypotenuse length is a straightforward process: Square the lengths of both legs, add these values, and then find the square root of the sum to determine the length of the hypotenuse. For example, if the legs of a right triangle measure 4 and 3 units, the hypotenuse length is calculated as \(\sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\) units. It is essential for students to remember that the hypotenuse is always the side opposite the right angle and that the Pythagorean theorem only applies to right triangles.

Solving Quadratic Equations

The Pythagorean theorem often results in a quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). In the context of finding the hypotenuse length, the quadratic equation takes the specific form \(x^2 = a^2 + b^2\), where \(x\) represents the hypotenuse length.

To solve such an equation, students typically rearrange it to \(x^2 - (a^2 + b^2) = 0\) and then find the roots. When \(a\) and \(b\) are positive real numbers, the equation will have a unique positive real solution for the hypotenuse length, x. This is because the square root of a positive number is also positive in the context of a right triangle.

In our exercise example, \(c^2 = 4^2 + 3^2\) simplifies to \(c^2 = 25\), which is a quadratic equation. By taking square roots, \(c = \pm \sqrt{25}\), but since we are dealing with a length, we only consider the positive root, \(c = 5\). It's crucial for learners to be able to recognize the connection between the Pythagorean theorem and quadratic equations and to use this relationship to solve for missing sides in right triangles.