Problem 47 - Precalculus

Question

Find the area of each triangle with measures given. $$a=14,000, b=16,500, c=18,700$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 118,568,851 square units.

Step 1: Calculate the Semi-Perimeter

The semi-perimeter of a triangle is needed to use Heron's Formula. Calculate it using the formula \[ s = \frac{a + b + c}{2} \]where \(a\), \(b\), and \(c\) are the sides of the triangle. Substitute the values:\[ s = \frac{14,000 + 16,500 + 18,700}{2} \]

Step 2: Simplify the Semi-Perimeter

Continue the calculation for the semi-perimeter:\[ s = \frac{49,200}{2} = 24,600 \]So, the semi-perimeter \(s\) is 24,600.

Step 3: Apply Heron's Formula for Area

Use Heron's Formula to find the area of the triangle, given by:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute \(s = 24,600\), \(a = 14,000\), \(b = 16,500\), and \(c = 18,700\) into the equation.

Step 4: Calculate Terms in Heron's Formula

Find each term:\[ s-a = 24,600 - 14,000 = 10,600 \]\[ s-b = 24,600 - 16,500 = 8,100 \]\[ s-c = 24,600 - 18,700 = 5,900 \]

Step 5: Compute the Area

Substitute the calculated terms back into Heron's Formula:\[ A = \sqrt{24,600 \times 10,600 \times 8,100 \times 5,900} \]Compute the multiplication inside the square root and then take the square root to find the area.

Key Concepts

Semi-Perimeter

In geometry, the semi-perimeter of a triangle, denoted as \(s\), is a key concept when working with Heron's Formula for determining the area of a triangle. The term "semi-perimeter" refers to half of the perimeter of the triangle. Calculating it involves adding up the lengths of all three sides of the triangle and then dividing by two. This value of \(s\) is fundamental because it simplifies the formula and calculations involved in finding the area.

The formula to find the semi-perimeter is:

• \( s = \frac{a + b + c}{2} \)

where \(a\), \(b\), and \(c\) are the side lengths of the triangle.

In our exercise, the semi-perimeter is calculated as follows:

• Add the side lengths: \(14,000 + 16,500 + 18,700 = 49,200\)
• Divide the total by 2: \(s = \frac{49,200}{2} = 24,600\)

This result serves as a foundation to apply Heron's formula, allowing for the calculation of the triangle's area.

Triangle Area

The area of a triangle is a measure of the space enclosed within its three sides. Heron's Formula is a powerful tool used to calculate this area when you know the lengths of all three sides of the triangle. This formula is particularly useful because it doesn't require knowledge of angles or any additional heights, just the sides.

Heron's Formula is expressed as:

• \( A = \sqrt{s(s-a)(s-b)(s-c)} \)

where:

• \(A\) is the area of the triangle,
• \(s\) is the semi-perimeter,
• \(a\), \(b\), and \(c\) are the side lengths.

By plugging in the semi-perimeter and the side lengths, we first calculate the individual differences:

• \(s-a = 24,600 - 14,000 = 10,600\)
• \(s-b = 24,600 - 16,500 = 8,100\)
• \(s-c = 24,600 - 18,700 = 5,900\)

Once these values are determined, they are substituted back to Heron's formula to find the area by calculating the product, taking the square root, and arriving at the desired measure of space.

Triangle Side Lengths

Triangle side lengths play a crucial role in determining various properties of a triangle, including its perimeter, semi-perimeter, and ultimately its area using Heron's Formula. In any triangle, these side lengths are the linear distances measured between each pair of vertices, and they must satisfy the triangle inequality theorem (i.e., the sum of the lengths of any two sides must be greater than the length of the third side).

Let's discuss the given side lengths in the exercise:

• Side \(a\) = 14,000
• Side \(b\) = 16,500
• Side \(c\) = 18,700

Using these side lengths with \(s\) calculated as 24,600, they enable us to apply Heron's Formula effectively. Moreover, their measurements in the problem statement already comply with the conditions set by the triangle inequality theorem, ensuring the existence of a valid triangle.

Knowing the length of each side also allows for further calculations beyond area, such as exploring angles using other trigonometric identities or determining if the triangle is right, acute, or obtuse by the relationship of its sides.