Problem 33 - College Algebra with Modeling and Visualization

Question

Find the area of the right triangle that satisfies the conditions. Approximate values to the nearest tenth when appropriate. Legs with lengths 3 feet and 6 feet

Step-by-Step Solution

Answer

The area is 9 square feet.

Step 1: Identify the Triangle Type

Given a right triangle with legs of lengths 3 feet and 6 feet. In a right triangle, the two legs are perpendicular to each other.

Step 2: Use the Area Formula for a Right Triangle

The area of a right triangle can be found using the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Here, we use the lengths of the legs as the base and height.

Step 3: Substitute the Values

Substitute the lengths of the legs into the formula: \( \text{Area} = \frac{1}{2} \times 3 \times 6 \).

Step 4: Calculate the Area

Perform the multiplication: \( \frac{1}{2} \times 3 \times 6 = \frac{1}{2} \times 18 = 9 \). Thus, the area of the triangle is 9 square feet.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This angle is known as the right angle. The two sides forming this right angle are called the 'legs' of the triangle, while the side opposite the right angle is the 'hypotenuse'. Right triangles are important in geometry because they serve as the basis for the Pythagorean theorem and trigonometric ratios.
Recognizing a right triangle is straightforward. Look for:

• One angle of 90 degrees (right angle)
• Two perpendicular legs
• One hypotenuse (the longest side)

Understanding these basic features helps solve many geometric problems involving right triangles.

Area Formula

To find the area of a right triangle, you can use a simple but powerful formula. The area formula for any triangle is \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]For right triangles, the two legs can be directly used as the base and the height because they are perpendicular. This makes calculations straightforward and accurate, without needing to make any additional conversions or assumptions.
The simplicity of the right triangle area formula often helps in quickly solving problems without unnecessary steps, ensuring both effectiveness and efficiency.

Geometry Concepts

Geometry concepts apply practically in numerous situations, from construction to art. Right triangles, especially, are foundational in these applications due to their predictable properties. Understanding basic geometry concepts like perpendicular lines and symmetry is crucial for tackling more complex problems.
In the case of our exercise, the problem gave us the legs of the triangle:

• Leg 1: 3 feet
• Leg 2: 6 feet

This straightforward information allows for direct application of the area formula, illustrating how geometry simplifies even seemingly complex situations.

Solving Problems with Steps

Approaching problems step-by-step enhances understanding and accuracy. Let's break down the solution process for finding the area of a right triangle as depicted in the exercise:

• **Identify** the type of triangle and its parts.
• **Use** the appropriate geometric formula (here, the area formula for right triangles).
• **Substitute** the known values (the lengths of the legs) into the formula.
• **Calculate** to find the solution.

This clear, structured approach not only makes problems easier to solve but also builds confidence in applying mathematical concepts in various scenarios. Practicing these steps ensures that they become second nature, making future problems less intimidating.