Problem 32 - 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. \(b=1.2\) miles, \(c=2\) miles

Step-by-Step Solution

Answer

The missing side \(a\) is 1.6 miles, and the perimeter is 4.8 miles.

Step 1: Understand Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), the equation is \(a^2 + b^2 = c^2\). We need to find the length of \(a\) given \(b = 1.2\) and \(c = 2\).

Step 2: Substitute Known Values into the Equation

Substitute \(b = 1.2\) miles and \(c = 2\) miles into the Pythagorean equation. This gives us: \(a^2 + (1.2)^2 = (2)^2\).

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

Calculate \(b^2\) as follows: \(1.2^2 = 1.44\). Calculate \(c^2\) as follows: \(2^2 = 4\).

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

Re-arrange the equation to find \(a^2\): \(a^2 + 1.44 = 4\). Subtract 1.44 from both sides to get \(a^2 = 4 - 1.44 = 2.56\).

Step 5: Find \(a\) by Taking the Square Root

To find the value of \(a\), take the square root of \(2.56\): \(a = \sqrt{2.56} = 1.6\). The value of \(a\) is 1.6 miles.

Step 6: Calculate the Perimeter

The perimeter \(P\) of the triangle is the sum of all three sides: \(P = a + b + c\). Substituting \(a = 1.6\), \(b = 1.2\), and \(c = 2\), we have \(P = 1.6 + 1.2 + 2 = 4.8\) miles.

Key Concepts

Right Triangle

A right triangle is a type of triangle that features one angle precisely equal to 90 degrees. This specific angle forms a corner known as the right angle. In a right triangle, the sides have special designations. The two sides at the right angle are referred to as the 'legs', and these are typically labeled as sides \(a\) and \(b\). The side opposite the right angle is called the 'hypotenuse', and it is labeled as \(c\). This setup allows for the application of the Pythagorean Theorem, which solves for missing side lengths when any two sides are known. Understanding the basic structure of a right triangle helps in many applications of geometry, particularly because it provides a foundational understanding for calculating distances and angles.

Hypotenuse

The hypotenuse is the longest side in a right triangle and is always located opposite the right angle. Its significance goes beyond merely being the longest side. It plays a crucial role in the Pythagorean Theorem, which is a cornerstone for many calculations in geometry. This theorem indicates that the square of the hypotenuse's length (\(c^2\)) is equal to the sum of the squares of the other two sides \(a^2\) and \(b^2\). Thus, if you know the lengths of the legs, you can calculate the hypotenuse. Conversely, knowing one leg and the hypotenuse allows you to find the other leg, as demonstrated in our example. For right triangles where \(b = 1.2\) miles and \(c = 2\) miles, knowing these values lets you calculate the missing side, \(a\), efficiently using this relationship.

Perimeter Calculation

The perimeter of a right triangle is simply the total length of all its sides added together. This includes the two legs and the hypotenuse. To calculate the perimeter, you use the formula: \[ P = a + b + c \] where \(a\), \(b\), and \(c\) represent the lengths of the sides. For instance, using our calculated values of \(a = 1.6\) miles, \(b = 1.2\) miles, and \(c = 2\) miles, we find the perimeter by substituting these values into our formula. In this example, the perimeter calculation shows: \[ P = 1.6 + 1.2 + 2 = 4.8 \] This straightforward method clearly shows how the concept of perimeter applies to right triangles and is crucial for finding the total boundary measurement of such geometric shapes.