Problem 77 - Algebra for College Students

Question

Find, to the nearest square centimeter, the area of a triangle that measures 14 centimeters by 16 centimeters by 18 centimeters.

Step-by-Step Solution

Answer

The area of the triangle is 107 cm².

Step 1: Understand Heron's Formula

To find the area of a triangle when all sides are known, we use Heron's formula. This formula is especially useful for triangles where the height is not readily available. Heron's formula is \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( a, b, \) and \( c \) are the lengths of the sides of the triangle, and \( s = \frac{a+b+c}{2} \) is the semi-perimeter of the triangle.

Step 2: Calculate the Semi-Perimeter

First, we need to calculate the semi-perimeter \( s \) of the triangle. For a triangle with side lengths 14 cm, 16 cm, and 18 cm, the semi-perimeter \( s \) is calculated as follows: \[ s = \frac{14+16+18}{2} = 24 \text{ cm} \]

Step 3: Plug Values into Heron's Formula

Now, substitute the values you have into Heron's Formula:\[ A = \sqrt{24(24-14)(24-16)(24-18)} \]This equation accounts for each side length and the respective differences from the semi-perimeter.

Step 4: Simplify Inside the Square Root

Continue by simplifying inside the square root:\[ A = \sqrt{24 \times 10 \times 8 \times 6} \]This step involves basic arithmetic operation of subtraction and multiplication.

Step 5: Calculate the Product

Next, calculate the product of the numbers under the square root:\[ 24 \times 10 \times 8 \times 6 = 11520 \]

Step 6: Find the Square Root

Finally, find the square root of the calculated product to obtain the area:\[ A = \sqrt{11520} \approx 107.31 \text{ cm}^2 \]Upon rounding to the nearest square centimeter, the area of the triangle is 107 cm².

Key Concepts

Heron's Formula

When you know all three sides of a triangle, but not the height, Heron's Formula offers a great method to find the area. It is particularly useful for triangles that are not right-angled or when you don't know the height. The formula is expressed as \( A = \sqrt{s(s-a)(s-b)(s-c)} \). Here, \( a, b, \) and \( c \) represent the lengths of each side of the triangle. The variable \( s \) denotes the semi-perimeter, which is half of the triangle's perimeter. Heron's Formula is a vital tool in geometry, providing a simple way to deduce the area from limited information.

Semi-Perimeter

Understanding the semi-perimeter is crucial for applying Heron's Formula. The semi-perimeter \( s \) is calculated by adding the lengths of all triangle sides and dividing by two. For example, if a triangle has sides measuring 14 cm, 16 cm, and 18 cm, its semi-perimeter is calculated as \( s = \frac{14+16+18}{2} = 24\) cm. The semi-perimeter serves as a central value around which the formula revolves, making it an essential step in the calculation process.

Square Root Calculation

Once the product under the square root has been determined, finding the square root itself is the next step. In this particular problem, the square root calculation involves \( \sqrt{11520} \). Calculating the square root can be done using a calculator. It often involves finding a number that when multiplied by itself gives the original value. Here, \( \sqrt{11520} \approx 107.31 \). This step transforms our previous work into the area measurement, making conversion of numerical data into usable geometric knowledge.

Geometry Problem-Solving

Solving geometry problems like this one involves understanding the relationship between sides, angles, and other dimensions of shapes. This exercise demonstrated a step-by-step approach in calculating a triangle's area, even without knowing the height. By following a structured method, such as:

• Using known formulas
• Breaking down larger calculations into smaller, manageable parts
• Doing arithmetic carefully

Geometry problem-solving enhances your mathematical reasoning and provides practical skills applicable in numerous fields.