Problem 27 - College Algebra with Modeling and Visualization

Question

Use the Pythagorean theorem to find the missing side of the right triangle with legs a and \(b\) and hypotenuse \(c .\) Then calculate the perimeter. Approximate values to the nearest tenth when appropriate. \(a=60\) feet, \(b=11\) feet

Step-by-Step Solution

Answer

The hypotenuse is 61 feet, and the perimeter is 132 feet.

Step 1: Identify Given Values

We are given that the two legs of the right triangle are \( a = 60 \) feet and \( b = 11 \) feet. These are the two shorter sides of the triangle, and we need to find the hypotenuse \( c \).

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem states that \( a^2 + b^2 = c^2 \). Substitute the known values into this equation: \( 60^2 + 11^2 = c^2 \).

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

Calculate each square: \( 60^2 = 3600 \) and \( 11^2 = 121 \).

Step 4: Sum the Squares

Add the squares together: \( 3600 + 121 = 3721 \).

Step 5: Solve for \( c \)

Take the square root of both sides to solve for \( c \): \( c = \sqrt{3721} = 61 \).

Step 6: Calculate the Perimeter

The perimeter of the triangle is the sum of all the sides: \( a + b + c = 60 + 11 + 61 = 132 \) feet.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle exactly equal to 90 degrees. This is known as a right angle. In every right triangle, there are two shorter sides called "legs" and one longer side known as the "hypotenuse." The right angle is formed where the two legs meet. Drawing the triangle can help you visualize where each side is. If you label the triangle with the right angle at the bottom left corner:

• The horizontal leg is your base.
• The vertical leg stands upright from the base.
• The hypotenuse extends diagonally, connecting the open ends of the legs.

Remember, the unique property of right triangles is their adherence to the Pythagorean theorem, which links the lengths of the sides in an interesting way.

Hypotenuse

The hypotenuse is the longest side of a right triangle, because it is opposite the right angle. In our exercise, the hypotenuse is labeled as "c," which we need to find. It serves as the 'anchor' that holds the diagonal over the right angle. To calculate the hypotenuse when the other two sides (legs) are known, you use the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] This formula tells you that the sum of the squares of the two legs equals the square of the hypotenuse. Plug in the given lengths of the legs into the equation: \[ 60^2 + 11^2 = c^2 \] Calculate to find: \( 3600 + 121 = 3721 \) Finally, find the length of the hypotenuse by taking the square root of 3721, which results in 61 feet. Now, you have the length of the hypotenuse, completing the triangle's side measurements.

Perimeter

The perimeter of a triangle is the total distance around the shape. It is simply the sum of the lengths of all three sides. In the case of our right triangle exercise, after finding all sides:

• Leg one (\(a\)): 60 feet
• Leg two (\(b\)): 11 feet
• Hypotenuse (\(c\)): 61 feet

The perimeter formula is:\[ \text{Perimeter} = a + b + c \]So, substitute our known side lengths into the formula:\[ 60 + 11 + 61 = 132 \] By adding these, you get a perimeter of 132 feet. Calculating the perimeter is a straightforward process once all the sides are known, and it provides a complete picture of the boundary length of the triangle.