Problem 29 - Precalculus Mathematics for Calculus

Question

Find the area of the triangle whose sides have the given lengths. \(a=9, \quad b=12, \quad c=15\)

Step-by-Step Solution

Answer

The area of the triangle is 54 square units.

Step 1: Calculate the Semi-Perimeter

The semi-perimeter of a triangle is given by the formula \( s = \frac{a+b+c}{2} \). Substitute the given side lengths to find \( s \).\[ s = \frac{9+12+15}{2} = \frac{36}{2} = 18 \]

Step 2: Apply Heron's Formula

Heron's Formula allows us to find the area of a triangle given its side lengths. The formula is:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \( s \) is the semi-perimeter calculated in Step 1.

Step 3: Substitute the Values into Heron's Formula

Using the values for \( s \), \( a \), \( b \), and \( c \):\[ A = \sqrt{18(18-9)(18-12)(18-15)} \] Simplify inside the square root:\[ A = \sqrt{18 \times 9 \times 6 \times 3} \]

Step 4: Simplify the Expression

Calculate the value inside the square root:\[ A = \sqrt{18 \times 9 \times 6 \times 3} = \sqrt{2916} \]

Step 5: Compute the Final Area

Calculate the square root:\[ A = \sqrt{2916} = 54 \] Therefore, the area of the triangle is 54 square units.

Key Concepts

Semi-Perimeter

The semi-perimeter of a triangle is an important concept in triangle geometry. The semi-perimeter is half the perimeter of the triangle. When you have the side lengths of a triangle, you can calculate the semi-perimeter using the formula:

• \( s = \frac{a+b+c}{2} \)

To find our semi-perimeter, simply add all the side lengths and divide by 2. For instance, in the problem with sides 9, 12, and 15, you add these together to get 36. Then divide by 2 to find the semi-perimeter:
\( s = \frac{36}{2} = 18 \)
This step is essential for using Heron's Formula to find the area.

Triangle Area Calculation

Calculating the area of a triangle can be tricky if you only have the side lengths. However, Heron's Formula provides a way to do this using the semi-perimeter. Once you have the semi-perimeter, the formula for the area \( A \) of the triangle is:

• \( A = \sqrt{s(s-a)(s-b)(s-c)} \)

Let's break it down:

• The semi-perimeter \( s \) is found first.
• Subtract each side length from the semi-perimeter.
• Multiply those results together.
• Take the square root at the end.

This requires some careful calculations, but it's a method that works universally for any triangle.

Geometry Problem-Solving

Solving geometry problems isn’t just about crunching numbers; it’s about understanding the relationships between geometric figures. For triangles, especially, there are useful relationships and formulas. Tackling these problems efficiently means:

• Knowing the necessary formulas like Heron's and semi-perimeter calculation.
• Breaking the problem into smaller steps to avoid mistakes.
• Understanding how different elements of the triangle, like sides and angles, relate to each other.

In our case, knowing how to calculate the semi-perimeter directly affects how we can calculate the area using Heron's formula.
Enjoying the problem-solving process is key to mastering geometry!

Mathematical Formulas

Mathematical formulas serve as tools that allow us to solve problems across various topics, including geometry. Heron’s Formula is a classic example in the realm of triangles, helping us solve for areas without needing height:

• First, calculate the semi-perimeter.
• Apply Heron's Formula to find the area directly.

Other important formulas can include the Pythagorean Theorem or the formula for the circumference of a circle. Becoming familiar with different mathematical formulas helps:

• Solve complex problems with ease.
• Validate your understanding by arriving at consistent results.
• Develop a toolkit for different mathematical scenarios.

Knowing the right formula to use is a crucial skill in math and keeps problem-solving effective and efficient.