Problem 19 - Elementary and Intermediate Algebra

Question

The lengths of two sides of the right triangle \(A B C\) are given. Find the length of the missing side. \(b=9 \mathrm{ft}\) and \(c=41 \mathrm{ft}\) (RIGHT TRIANGLE CAN'T COPY)

Step-by-Step Solution

Answer

The length of the missing side \(a\) is 40 ft.

Step 1: Identify Known Values

In the right triangle ABC, you're given side \(b = 9\, \mathrm{ft}\) and the hypotenuse \(c = 41\, \mathrm{ft}\). The missing side is \(a\), which we'll find using the Pythagorean theorem.

Step 2: Recall Pythagorean Theorem

The Pythagorean theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), the equation is: \[a^2 + b^2 = c^2\]. We will use this equation to solve for \(a\).

Step 3: Substitute the Known Values

Substitute the known values of \(b\) and \(c\) into the Pythagorean theorem: \[a^2 + 9^2 = 41^2\].

Step 4: Simplify and Solve for \(a^2\)

Calculate \(9^2\) and \(41^2\): \[a^2 + 81 = 1681\]. Solve for \(a^2\) by subtracting 81 from both sides: \[a^2 = 1681 - 81\] , which simplifies to \(a^2 = 1600\).

Step 5: Solve for \(a\)

Find \(a\) by taking the square root of \(a^2\): \[a = \sqrt{1600}\]. Calculate the square root: \(a = 40\).

Key Concepts

Right Triangle

A right triangle is a specific type of triangle that has one angle measuring exactly 90 degrees. This characteristic makes right triangles unique and valuable when solving for unknown side lengths. The right angle is often symbolized by a small square at the vertex of the triangle. In a right triangle, one can use the Pythagorean theorem, a helpful mathematical principle, to find the relationship between the sides.
While dealing with right triangles, it's important to remember:

• The side opposite to the right angle is the longest and is termed the hypotenuse.
• The two other sides are referred to as the legs of the triangle.
• The right angle is always pivotal in setting up the Pythagorean theorem.

Understanding these concepts allows us to accurately apply the theorem to solve for missing side lengths in any right triangle.

Hypotenuse

In a right triangle, the hypotenuse is the side opposite the right angle. It holds the longest length among all three sides. Knowing the hypotenuse is crucial because it plays a key role when employing the Pythagorean theorem.
To identify the hypotenuse in a right triangle:

• Look for the side across from the 90-degree angle.
• It is the diagonal side when imagined in standard orientation.
• In the context of the Pythagorean theorem, it is always represented by 'c'.

The hypotenuse is necessary for calculating unknown sides as it is always included in forming the equation: \(a^2 + b^2 = c^2\). This formula forms the foundation of using the Pythagorean theorem effectively.

Legs of Triangle

The legs of a right triangle are the two sides that form the right angle. They are shorter than the hypotenuse and are labeled as side 'a' and side 'b' in formulas. These legs work together with the hypotenuse in the Pythagorean theorem to establish a mathematical relationship between the sides.
It's important to note:

• The legs are perpendicular to each other.
• They form the sides of the right angle.
• You can use them interchangeably in the Pythagorean theorem as long as correct values are substituted.

This consistent interrelation allows us to find an unknown leg length when the hypotenuse and one leg are known, displaying the versatility of the Pythagorean theorem.