Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Find the area of each triangle. \(a=154\) centimeters, \(b=179\) centimeters. \(c=183\) centimeters

Step-by-Step Solution

Answer

The area of the triangle is approximately 12,637.5 cm².

Step 1: Calculate the Semiperimeter

To use Heron's formula to find the area of the triangle, first calculate the semiperimeter \( s \) of the triangle using the formula: \( s = \frac{a + b + c}{2} \). Here, \( a = 154 \), \( b = 179 \), and \( c = 183 \). So, \[ s = \frac{154 + 179 + 183}{2} = \frac{516}{2} = 258 \text{ cm} \].

Step 2: Use Heron's Formula for Area

Heron's formula for the area \( A \) of a triangle is \( A = \sqrt{s(s-a)(s-b)(s-c)} \). Substitute the values of \( s \), \( a \), \( b \), and \( c \) into the formula: \( A = \sqrt{258(258-154)(258-179)(258-183)} \).

Step 3: Calculate the Individual Terms

Calculate the terms inside the square root separately: \( s-a = 258 - 154 = 104 \), \( s-b = 258 - 179 = 79 \), and \( s-c = 258 - 183 = 75 \).

Step 4: Compute the Product Under the Square Root

Now compute the product: \[ 258 \times 104 \times 79 \times 75 = 159785400 \].

Step 5: Calculate the Square Root

Finally, take the square root of the product: \[ A = \sqrt{159785400} \approx 12637.5 \text{ cm}^2 \].

Step 6: Verify the Units

Make sure the units for area are correct. Since the side lengths are in centimeters, the area should be in square centimeters.

Key Concepts

Semiperimeter

Before calculating the area of a triangle using Heron's formula, you must find the semiperimeter. This is a crucial step that determines the starting point for further calculations. The semiperimeter, denoted as \( s \), of a triangle is essentially half of its perimeter. If you think about the perimeter as the total distance around the triangle, finding half of that gives you the semiperimeter.
To calculate it, you use the formula:

• \( s = \frac{a+b+c}{2} \)

Where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.
In our example, these side lengths are \( a = 154 \) cm, \( b = 179 \) cm, and \( c = 183 \) cm.
Plug them into the formula:

• \( s = \frac{154+179+183}{2} \)
• \( s = \frac{516}{2} \)
• \( s = 258 \) cm

Now that you have the semiperimeter, you're ready to tackle the area calculation using Heron's formula.

Triangle Area Calculation

Calculating the area of a triangle becomes straightforward once you know the semiperimeter. Heron's formula is a powerful tool for this purpose. It provides a way to calculate the area when you have the lengths of all three sides. The formula is:

• \( A = \sqrt{s(s-a)(s-b)(s-c)} \)

Where \( A \) is the area, \( s \) is the semiperimeter, and \( a \), \( b \), and \( c \) are the sides.
With the semiperimeter \( s = 258 \) cm already calculated, and sides \( a = 154 \) cm, \( b = 179 \) cm, and \( c = 183 \) cm, substitute these values into the formula:

• \( A = \sqrt{258(258-154)(258-179)(258-183)} \)

It's essential to break down the expression inside the square root to avoid errors, which demands calculating each term like:

• \( 258-154 = 104 \)
• \( 258-179 = 79 \)
• \( 258-183 = 75 \)

So the expression becomes:

• \( A = \sqrt{258 \times 104 \times 79 \times 75} \)

Solving this expression will give you the area of the triangle.

Square Root Calculation

After setting up Heron's formula for the area, the final challenge lies in executing the square root calculation. This involves calculating the product inside the square root, followed by finding the square root itself.
Let's start by computing the product of the terms previously found:

• \( 258 \times 104 \times 79 \times 75 = 159785400 \)

Once you have this large number, the task is to find the square root:

• \( A = \sqrt{159785400} \approx 12637.5 \text{ cm}^2 \)

Calculating the square root can be performed using a calculator for accuracy since manually doing this might be cumbersome for such a large number.
Remember, when dealing with units, if the side lengths are in centimeters, as in this problem, the area will naturally be in square centimeters.
This step completes the application of Heron's formula, giving you the triangle's area in squared units.