Problem 79 - 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. $$a=13, c=85 ; \text { find } b$$

Step-by-Step Solution

Answer

The length of side \(b\) is 84.

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. This can be written as the equation: \[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse.

Step 2: Substitute the known values into the equation

We substitute the given lengths \(a = 13\) and \(c = 85\) into the equation: \[ 13^2 + b^2 = 85^2 \].

Step 3: Simplify the squares in the equation

Calculate the squares of the known values: \[ 13^2 = 169 \] and \[ 85^2 = 7225 \]. Substituting these into the equation gives: \[ 169 + b^2 = 7225 \].

Step 4: Solve for \(b^2\)

Isolate \(b^2\) by subtracting 169 from both sides of the equation: \[ b^2 = 7225 - 169 \].

Step 5: Calculate the difference in the equation

Subtract 169 from 7225 to get \(b^2 = 7056\).

Step 6: Find the length of side \(b\)

Take the square root of both sides to solve for \(b\): \[ b = \sqrt{7056} \]. Calculate \(\sqrt{7056}\) to get \(b = 84\).

Key Concepts

Understanding Right Triangles

In geometry, a right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The 90-degree angle is also known as a right angle, making the triangle referred to as a 'right' triangle. This type of triangle is fundamental in trigonometry and geometry due to its unique properties that allow the use of the Pythagorean Theorem.

The right triangle consists of three sides:

• The side opposite the right angle is called the hypotenuse. This is the longest side in a right triangle.
• The other two sides are referred to as 'legs.'

Right triangles are everywhere, from construction and architecture to computer graphics. Understanding their properties is crucial for practical applications and other mathematics fields.

Comprehending the Hypotenuse

The hypotenuse is a critical concept when working with right triangles. In a right triangle, the hypotenuse is the side lying opposite the right angle and is the longest side of the triangle. This length is always greater than the lengths of either of the two legs. This property is vital when using the Pythagorean Theorem to solve problems related to right triangles.

Evaluate the importance of the hypotenuse:

• It plays a central role in solving for the unknown sides of a right triangle using the Pythagorean Theorem.
• Measurements and calculations involving hypotenuses are common in fields ranging from physics to engineering.

In our original exercise, you were given the hypotenuse's length (85), forming a base to calculate the unknown leg.

Solving Equations with the Pythagorean Theorem

Solving for the unknown side in a right triangle often involves the Pythagorean Theorem, which states that the sum of the squares of the two legs is equal to the square of the hypotenuse: \[ a^2 + b^2 = c^2 \].

To solve for an unknown side using this theorem, follow these steps:

• Substitute the known values into the equation. For instance, if given \(a = 13\) and \(c = 85\), substitute these to get \(13^2 + b^2 = 85^2\).
• Calculate the squares of the known sides; here, \(13^2 = 169\) and \(85^2 = 7225\).
• Rearrange the equation to isolate the unknown side, \(b^2 = 7225 - 169\).
• Calculate the difference, then take the square root to find \(b\). In this case, \(b = \sqrt{7056} = 84\).

Calculating each of these steps carefully ensures that you can efficiently solve for any unknown side of a right triangle, validating your understanding of basic algebra and geometry principles.