Problem 35 - 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. Hypotenuse 15 inches and leg 11 inches

Step-by-Step Solution

Answer

The missing side is approximately 10.2 inches, and the perimeter is 36.2 inches.

Step 1: Identify the Triangle's Sides

In the right triangle given, we are provided with the hypotenuse \( c = 15 \) inches and one leg \( a = 11 \) inches. We need to find the missing leg \( b \).

Step 2: Apply the Pythagorean Theorem

The Pythagorean theorem states \( a^2 + b^2 = c^2 \). Substituting the known values into the equation, we have:\[ 11^2 + b^2 = 15^2 \]

Step 3: Simplify the Equation

Calculate the square of the known values:\[ 121 + b^2 = 225 \]

Step 4: Solve for b²

Subtract \( 121 \) from \( 225 \) to isolate \( b^2 \):\[ b^2 = 225 - 121 \]\[ b^2 = 104 \]

Step 5: Calculate b

Take the square root of both sides to solve for \( b \):\[ b = \sqrt{104} \]\[ b \approx 10.2 \] (rounded to the nearest tenth)

Step 6: Calculate the Perimeter of the Triangle

The perimeter \( P \) is the sum of all sides:\[ P = a + b + c \]\[ P = 11 + 10.2 + 15 \]\[ P = 36.2 \] inches, approximated to the nearest tenth.

Key Concepts

Understanding a Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This particular angle is known as the right angle. In every right triangle, the sides have specific names based on their position relative to the right angle. The side opposite the right angle is the longest and is called the hypotenuse. The other two sides are the legs. Right triangles are unique because they allow us to use the Pythagorean theorem to find missing side lengths, a fundamental concept in geometry. Remember, the sum of the squares of the legs equals the square of the hypotenuse.

Discovering the Hypotenuse

The hypotenuse is the defining feature of a right triangle. It is always opposite the right angle and is longer than either of the other sides, the legs. This is because, in a right triangle, the hypotenuse is the longest side due to the Pythagorean theorem's principle. When you know the lengths of both legs, you can easily determine the hypotenuse using the formula: \( c = \sqrt{a^2 + b^2}\). In the provided example, the hypotenuse is already given as 15 inches, guiding us to find the missing side using our existing knowledge of the theorem.

Perimeter Calculation in Triangles

Perimeter calculation in a triangle is straightforward once you identify the lengths of all sides. The perimeter is simply the total distance around the triangle. In the case of a right triangle with legs measuring 11 inches and 10.2 inches, and a hypotenuse of 15 inches, you calculate the perimeter by summing these values.

• First leg: 11 inches
• Second leg: 10.2 inches
• Hypotenuse: 15 inches

The perimeter comes out to be 36.2 inches. This calculation is done by adding these three measurements together, offering a simple way to measure the sum of a triangle's borders.

Basic Concepts of Geometry

Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, and shapes. Understanding geometry allows us to depict the physical world in mathematical form and solve practical problems. Right triangles are a foundational concept in geometry, heavily utilized for calculations involving distances and spaces. Learning to use geometric principles, such as the Pythagorean theorem, helps in solving complex problems by breaking them down into simpler parts. By understanding how figures like right triangles work, students gain the ability to interpret and manipulate various shapes and forms critically.