Problem 29 - 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. \(a=5\) centimeters, \(c=13\) centimeters

Step-by-Step Solution

Answer

The missing side \( b \) is 12 cm, and the perimeter is 30 cm.

Step 1: Recall the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides \( a \) and \( b \). The formula is \( a^2 + b^2 = c^2 \).

Step 2: Substitute Known Values Into the Equation

Insert the given values into the formula: \( 5^2 + b^2 = 13^2 \). This becomes \( 25 + b^2 = 169 \).

Step 3: Solve for Unknown Side \(b\)

To find \( b \), subtract 25 from both sides: \( b^2 = 169 - 25 \). This simplifies to \( b^2 = 144 \). Take the square root of both sides to find \( b \): \( b = \sqrt{144} = 12 \).

Step 4: Calculate Perimeter of the Triangle

The perimeter \( P \) of a triangle is the sum of all its sides: \( P = a + b + c \). Substitute the values \( a = 5 \), \( b = 12 \), and \( c = 13 \): \( P = 5 + 12 + 13 \).

Step 5: Compute and Round if Necessary

Add the side lengths to find the perimeter: \( P = 30 \) centimeters. Since 30 is an integer, rounding to the nearest tenth is unnecessary.

Key Concepts

Right Triangle

Right triangles are a special type of triangle that contain one 90-degree angle, known as a right angle. This type of triangle is unique because the side opposite the right angle is called the hypotenuse and is the longest side of the triangle. The other two sides are referred to as the legs. In every right triangle, the relationships between the lengths of the legs and the hypotenuse are governed by the Pythagorean Theorem.

• One angle is always 90 degrees.
• The hypotenuse is always the side opposite the right angle.
• Right triangles are extremely useful in various fields such as geometry, physics, engineering, and architecture.

Understanding the properties and relationship between the sides of a right triangle is essential for solving many practical problems, including this exercise.

Hypotenuse

The hypotenuse plays a crucial role in a right triangle. It is always opposite the right angle. According to the Pythagorean Theorem, the hypotenuse's length can be determined if the lengths of the other two sides are known. The theorem is expressed in the formula:\[a^2 + b^2 = c^2\]where \(c\) is the hypotenuse and \(a\) and \(b\) are the lengths of the other sides.

This theorem allows for the calculations of unknown side lengths in right triangles by plugging in the known values. In our specific problem setting, given \(a = 5\) and \(c = 13\), we solve the equation to find the missing leg \(b\).

Given:

• \(a = 5\)
• \(c = 13\)

By simplifying:\[5^2 + b^2 = 13^2\]\[25 + b^2 = 169\]\[b^2 = 169 - 25 = 144\]Taking the square root:\[b = \sqrt{144} = 12\]Thus, understanding and using the hypotenuse and the Pythagorean Theorem is straightforward and vital in solving such problems.

Perimeter Calculation

Calculating the perimeter of a right triangle is a straightforward process after determining the lengths of all sides. The perimeter is essentially the sum of all these side lengths. Once you've found each side, adding them together will give you the perimeter.

In this exercise, after identifying that the sides of the triangle are \(a = 5\), \(b = 12\), and \(c = 13\), we can compute the perimeter using the formula:\[P = a + b + c\]Substitute the known values:\[P = 5 + 12 + 13\]This will result in:\[P = 30 \, \text{centimeters}\]There is no need to round 30 to the nearest tenth, since it's already an integer. By understanding how to find the perimeter, you can efficiently determine how much material you might need to enclose or highlight a triangle's boundary in real-world contexts such as crafting or construction.