Problem 82 - 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. $$a=9, b=10 ; \text { find } c$$

Step-by-Step Solution

Answer

The length of the hypotenuse \( c \) is approximately 13.45.

Step 1: Review the Pythagorean Theorem

For a right triangle, the Pythagorean Theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b) of the triangle. It is mathematically represented as \( c^2 = a^2 + b^2 \).

Step 2: Substitute the Known Values

In this exercise, we are given \( a = 9 \) and \( b = 10 \). Substitute these values into the equation \( c^2 = a^2 + b^2 \). This will give:\[ c^2 = 9^2 + 10^2 \].

Step 3: Perform the Squaring Operations

Calculate \( 9^2 \) and \( 10^2 \):- \( 9^2 = 81 \)- \( 10^2 = 100 \).

Step 4: Sum the Squares

Add the results from Step 3 together to find \( c^2 \):\[ c^2 = 81 + 100 = 181 \].

Step 5: Calculate the Length of the Hypotenuse

To find \( c \), take the square root of \( 181 \):\( c = \sqrt{181} \).Calculate \( \sqrt{181} \) which is approximately 13.45.

Key Concepts

Right Triangle

In geometry, a right triangle is a type of triangle that has one angle exactly 90 degrees. This distinct feature makes it unique compared to other triangles. The 90-degree angle is also known as a right angle. In a right triangle, the side opposite this right angle is always the longest side and is referred to as the hypotenuse. The other two sides adjacent to the right angle are known as the legs.
A right triangle can be very useful because it allows us to use the Pythagorean Theorem to calculate the lengths of the sides when some values are known. This theorem is applicable exclusively to right triangles, making it a key concept in geometry and various practical applications like construction and navigation.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is located opposite the right angle. One important characteristic is that it represents the diagonal in the right triangle. Its length is critical in solving problems involving right triangles by using the Pythagorean Theorem, which expresses the relationship between the hypotenuse and the other two sides, called legs.
The formula for the hypotenuse in a right triangle, according to the Pythagorean Theorem, is represented as follows:

• \(c^2 = a^2 + b^2\)
• Where \(c\) is the hypotenuse, and \(a\) and \(b\) are the two other sides.

In our case, if we have a right triangle with legs of lengths 9 and 10, this formula lets us find the hypotenuse. First, square the lengths of the legs, add those squares together, and then take the square root of that sum. This will give the value of the hypotenuse.

Square Root

The square root is a mathematical concept that helps us find a number which, when multiplied by itself, results in a given number. It is represented by the symbol \sqrt{}\. Calculating the square root is essential in the Pythagorean Theorem for solving the length of the hypotenuse in a right triangle.
For example, in the problem we're solving, we find \(c^2 = 181\). To solve for \(c\), the hypotenuse, we need to take the square root of 181. \(\sqrt{181}\) approximates to 13.45, showing us the length of the hypotenuse. Remember, understanding square roots can help greatly when working through many algebraic and geometric problems, particularly those involving right triangles.