Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Heron's Area Formula to find the area of the triangle. $$a=12, \quad b=17, \quad c=8$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 47.67 square units. This exact answer will depend on the precision of your calculator or software used.

Step 1: Calculation of Semi-perimeter

Use the given side lengths of the triangle to calculate the semi-perimeter 's'. This can be achieved using the formula \( s = \frac{a + b + c}{2} \). Hence, \( s = \frac{12 + 17 + 8}{2} = 18.5 \).

Step 2: Calculation of Area

Substitute 's', 'a', 'b', 'c' into Heron's formula to find the area of the triangle. The formula for the area according to Heron's formula is \( \sqrt{s(s - a)(s - b)(s - c)} \). Substitute the values into the formula to get \( \sqrt{18.5(18.5 - 12)(18.5 - 17)(18.5 - 8)} \).

Step 3: Solve the equation for Area

Solve the equation above to find the actual numerical value of the area. This should give us the area in square units of the triangle.

Key Concepts

Understanding Triangles

A triangle is a polygon with three edges and three vertices. It's one of the basic shapes in geometry and comes in various forms, such as equilateral, isosceles, and scalene. Each type depends on the lengths of its sides and the measures of its angles.
When working with triangles, it's vital to know the lengths of all three sides and understand the relationships between those sides and the triangle's angles. This knowledge can help in various mathematical scenarios, like determining the area of the triangle using various formulas.
The sides of a triangle are typically labeled as 'a', 'b', and 'c', with each side opposite an angle. The most interesting fact about triangles is that the sum of their internal angles always equals 180 degrees, which can help solve different triangle problems.
In our case, we are dealing with a triangle with sides of lengths 12, 17, and 8. To find the area of this triangle efficiently, we'll employ Heron's formula. Heron's formula is especially useful for triangles where the height is not known, which often happens in real-world applications.

Importance of the Semi-perimeter

The semi-perimeter of a triangle is half of its perimeter. It's a key component when using Heron's formula to calculate the area of a triangle.
To find the semi-perimeter, you add up all the side lengths and divide by two. For a triangle with sides 'a', 'b', and 'c', the equation is:

• \( s = \frac{a + b + c}{2} \)

This value is crucial because it allows us to break down and simplify the area calculation process. In our exercise, the semi-perimeter was calculated as 18.5.
Using the semi-perimeter is particularly useful in representing part of a computation involving multiple subtraction operations efficiently. This streamlines the follow-up processes in Heron's algorithm, setting a solid foundation for further calculations needed to solve the area of the triangle comprehensively.

Approach to Mathematical Problem Solving

Mathematical problem-solving is a skill set that bridges logical thinking and mathematical concepts to reach solutions. It involves understanding the problem, devising a plan, carrying out that plan, and finally evaluating the results.
In solving the problem of finding the area of a triangle using Heron's formula, we follow several steps:

• Firstly, clearly understand the problem by identifying the elements like the side lengths of the triangle.
• Next, strategize by identifying the proper formulas and mathematical principles involved, such as the semi-perimeter and Heron's formula.
• Execute the plan by substituting the known values into these formulas to carry out calculations.
• Lastly, review the results to ensure accuracy and correctness in the solution, confirming that all computations follow logically from the data provided.

Effective problem-solving in mathematics also entails checking your work carefully to ensure that the solution makes sense in the context of the problem. Following a structured approach helps eliminate errors and builds strong mathematical thinking skills.