Problem 44 - Precalculus

Question

Find the area of each triangle with measures given. $$a=40, b=50, c=60$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 984.375 square units.

Step 1: Understand the Problem

We need to find the area of a triangle given its side lengths. Here, the sides of the triangle are denoted by \(a = 40\), \(b = 50\), and \(c = 60\).

Step 2: Apply Heron's Formula

To find the area of a triangle when all sides are known, use Heron's formula. First, we calculate the semi-perimeter \(s\), which is half the sum of the sides: \(s = \frac{a + b + c}{2}\).

Step 3: Calculate the Semi-Perimeter

Compute the semi-perimeter \(s\): \[s = \frac{40 + 50 + 60}{2} = 75\]

Step 4: Use Heron's Formula for Area

Heron's formula states that the area \(A\) of the triangle is given by:\[A = \sqrt{s(s-a)(s-b)(s-c)}\]Substitute \(s = 75\), \(a = 40\), \(b = 50\), \(c = 60\).

Step 5: Calculate Individual Terms

Compute each term inside the square root:\(s-a = 75 - 40 = 35\)\(s-b = 75 - 50 = 25\)\(s-c = 75 - 60 = 15\)

Step 6: Compute the Area

Substitute the values into Heron's formula:\[A = \sqrt{75 \times 35 \times 25 \times 15}\]Calculate inside the square root:\[A = \sqrt{984375}\]Then find the numerical value of the square root.

Step 7: Final Simplification and Result

After calculating, we find:\[A \approx 984.375\ ext{square units}\]

Key Concepts

triangle area

Understanding how to calculate the area of a triangle is key in geometry. The area of a triangle can be found using different methods, depending on the information available. When you have the lengths of all three sides, Heron's Formula is your go-to tool. This formula allows you to find the area without knowing the height of the triangle. This is particularly useful in many mathematical problems and real-world applications. Heron's Formula requires you to first determine the semi-perimeter of the triangle, which we'll discuss next.
By applying Heron's Formula, you find the area by using:

• The three side lengths of the triangle
• The semi-perimeter, which is a crucial step in Heron's calculation

Each side of the triangle contributes to the area. Once calculated, the area gives you the region enclosed within the triangle's boundaries.

semi-perimeter

The semi-perimeter of a triangle is central to calculating the area using Heron's Formula. It represents half of the triangle's perimeter. To calculate it, you simply sum up all three side lengths and divide by two. For example, if a triangle has side lengths of 40, 50, and 60, the semi-perimeter is calculated by\[ s = \frac{40 + 50 + 60}{2} = 75 \]

This value is essential because it is used in the next step of Heron's Formula. Understanding the semi-perimeter is critical as it helps to break down the problem into manageable parts:

• Sum the side lengths: 40 + 50 + 60
• Divide by 2 to get the semi-perimeter: 75

This concept is not limited to just one application; it forms the foundational step necessary for accurately finding the area of the triangle using Heron's method.

triangle side lengths

The triangle's side lengths are the first pieces of information needed to utilize many formulas, including Heron's Formula. When given side lengths known as \(a\), \(b\), and \(c\), such as in this case \(a = 40\), \(b = 50\), and \(c = 60\), they allow us to compute not just the area but also the semi-perimeter.

These values not only define the triangle's shape but also are crucial to any further calculations:

• They provide the basis for calculating the semi-perimeter, \(s = \frac{a+b+c}{2}\)
• They determine each inside term in Heron's formula\((s-a)(s-b)(s-c)\)

Familiarity with the concept of triangle side lengths makes complex calculations manageable, as you see the step-by-step transformation of these basics into the meaningful concept of area.