Problem 23 - Algebra and Trigonometry

Question

Find the area of each triangle. Round answers to two decimal places. $$a=12, \quad b=35, \quad c=37$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 210 square units.

Step 1: Identify the formula

To find the area of a triangle with sides given but without height, use Heron's formula. Heron's formula for the area \(A\) of a triangle with sides \(a, b, c\) is: \[A = \sqrt{s(s-a)(s-b)(s-c)}\] where \(s\) is the semi-perimeter of the triangle. The semi-perimeter \(s\) is calculated as: \[s = \frac{a+b+c}{2}\]

Step 2: Calculate the semi-perimeter

Use the values of the sides to calculate the semi-perimeter: \[s = \frac{12 + 35 + 37}{2} = \frac{84}{2} = 42\]

Step 3: Substitute values into Heron's formula

Substitute the values of \(a, b, c\), and \(s\) into Heron's formula: \[A = \sqrt{42(42-12)(42-35)(42-37)} \] This simplifies to: \[A = \sqrt{42 \cdot 30 \cdot 7 \cdot 5}\]

Step 4: Compute the product inside the square root

First, compute the product inside the square root: \[42 \cdot 30 \cdot 7 \cdot 5 = 44100\]

Step 5: Take the square root

Take the square root of 44100 to find the area: \[A = \sqrt{44100} \approx 210\]

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle can be straightforward when you have the height and the base. You simply use the formula for the area of a triangle: \[\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}\]. However, this can get tricky when the height is not known. Fortunately, Heron's formula allows us to calculate the area using only the sides of the triangle. To use Heron's formula, we follow several steps. First, we calculate the semi-perimeter of the triangle. Next, we use the semi-perimeter along with the lengths of the sides to find the area through Heron's formula. This method is handy because it doesn't require knowledge of the triangle's height.

Semi-Perimeter of Triangle

The semi-perimeter of a triangle is half of its perimeter. This quantity is useful in Heron's formula and is represented by the letter 's.' To calculate it, add up the lengths of all three sides of the triangle and divide by two. For a triangle with sides of lengths \(a\), \(b\), and \(c\), the semi-perimeter is given by: \[s = \frac{a + b + c}{2}\] In our example, with sides 12, 35, and 37, the semi-perimeter becomes: \[s = \frac{12 + 35 + 37}{2} = 42\] This step is essential, as the semi-perimeter is used directly in the next steps of the Heron's formula calculation.

Square Root

Finding the square root of a number is a crucial part of using Heron's formula. The square root is essentially the value that, when multiplied by itself, gives the original number. In our final step, we need to calculate the square root of the result obtained from our intermediate multiplication. In the example problem, after substituting the semi-perimeter and side lengths into Heron's formula, we find: \[A = \sqrt{42(42-12)(42-35)(42-37)} = \sqrt{44100}\] Finally, taking the square root of 44100, we get: \[A = 210\] This means the area of the triangle is 210 square units. Round to the necessary precision as required. For most calculations, modern calculators can quickly provide the square root, making this step more manageable.