Problem 220 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. A triangular window has base eight feet and height six feet. What is its area?

Step-by-Step Solution

Answer

The area of the triangular window is 24 square feet.

Step 1 - Identify the formula for the area of a triangle

The formula for the area of a triangle is given by \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]

Step 2 - Insert the given values

Substitute the given base and height into the formula. The base is 8 feet and the height is 6 feet. Thus, the substitution will be: \[\text{Area} = \frac{1}{2} \times 8 \text{ feet} \times 6 \text{ feet}\]

Step 3 - Calculate the product within the formula

Perform the multiplication inside the formula: \[\text{Area} = \frac{1}{2} \times 48 \text{ square feet} = 24 \text{ square feet}\]

Step 4 - Simplify to get the final area

After performing the multiplication, simplify the result to get the final area. The area of the triangular window is 24 square feet.

Key Concepts

Area of a Triangle

Calculating the area of a triangle is an essential skill in geometry. A triangle's area helps us understand its size and is used in various real-life scenarios. The formula for the area of a triangle is \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). This formula works because a triangle is essentially half of a rectangle. By multiplying the base (the bottom edge) by the height (the perpendicular distance from the base to the top vertex), we calculate the area of a rectangle, then we take half to get the area of the triangle.
For example, imagine we have a triangular window with a base of 8 feet and a height of 6 feet. By inserting these values into our formula: \(\text{Area} = \frac{1}{2} \times 8 \text{ feet} \times 6 \text{ feet}\), we see that it simplifies to \(\text{Area} = 24 \text{ square feet}\). This straightforward process makes it easy to find the area of any triangle as long as we know the base and height.

Geometry in Algebra

When we calculate areas in geometry, we often integrate concepts from algebra. This integration helps strengthen our mathematical understanding and problem-solving skills. To calculate the area of a triangle, we use multiplication and simplification – basic algebraic operations.
Consider our triangular window example. Here’s how algebra comes into play:

• We start with our geometric formula \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\).
• Next, we substitute the given numerical values: \(\text{Area} = \frac{1}{2} \times 8 \text{ feet} \times 6 \text{ feet}\).
• Then, we perform the multiplication inside the formula: \(\text{Area} = \frac{1}{2} \times 48 \text{ square feet}\).
• Finally, we simplify by taking half of 48: \(\text{Area} = 24 \text{ square feet}\).

Throughout this process, we use algebraic principles to manipulate the formula and solve for the area. This demonstrates how closely related geometry and algebra are in many mathematical problems.

Step-by-Step Solution

Breaking down the problem into steps makes it much easier to solve. Let’s go through the steps in detail for our triangular window with a base of 8 feet and height of 6 feet:

• Step 1: Identify the formula. Here, we use \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\).
• Step 2: Insert the given values. Our base is 8 feet, and the height is 6 feet, so we substitute these into the formula: \(\text{Area} = \frac{1}{2} \times 8 \text{ feet} \times 6 \text{ feet}\).
• Step 3: Calculate the product. Perform the multiplication inside: \(\text{Area} = \frac{1}{2} \times 48 \text{ square feet}\).
• Step 4: Simplify to get the final area. Take half of 48 to get: \(\text{Area} = 24 \text{ square feet}\).

By following each step methodically, you ensure accuracy and make the problem less intimidating. This step-by-step approach is beneficial not just for this problem but for a wide range of mathematical exercises. Remember, practice makes perfect!