Problem 13 - Basic Technical Mathematics with Calculus

Question

Find the area of each triangle. Right triangle with legs \(3.46 \mathrm{ft}\) and \(2.55 \mathrm{ft}\)

Step-by-Step Solution

Answer

The area of the triangle is 4.4115 square feet.

Step 1: Understand the Right Triangle

In a right triangle, one of the angles is exactly 90 degrees, and the two sides that form this right angle are called the 'legs' of the triangle. The third side is the hypotenuse.

Step 2: Recall the Formula for the Area

The area of a triangle is calculated using the formula: \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\] In a right triangle, the legs can be used as the base and height of the triangle.

Step 3: Identify the Legs

For this right triangle, the given legs are the measurements: 3.46 feet and 2.55 feet. These serve as both the base and the height in the area formula.

Step 4: Substitute the Values into the Formula

Substitute the given values of the legs into the area formula: \[\text{Area} = \frac{1}{2} \times 3.46 \times 2.55\]

Step 5: Perform the Multiplication

Calculate the product of the legs: \[3.46 \times 2.55 = 8.823\]

Step 6: Calculate the Area

Now, divide the product by 2 to find the area: \[\text{Area} = \frac{1}{2} \times 8.823 = 4.4115\] The area of the triangle is thus 4.4115 square feet.

Key Concepts

Right Triangle

A right triangle is one of the most widely known shapes in geometry. Its notable feature is that it contains a 90-degree angle. The other two angles in a right triangle will always add up to 90 degrees, because the sum of angles in any triangle is always 180 degrees.

In right triangles, the sides have special terms:

• Legs: These are the two sides that form the right angle.
• Hypotenuse: This is the side opposite the right angle, and it is the longest side of the triangle.

These triangles are useful because of the properties that relate the sides, allowing for easy calculations of area and other parameters.

Area Formula

The area of a triangle can be determined through a simple mathematical formula: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\\)
For right triangles, this formula becomes particularly straightforward because the two legs can be seen as the base and the height. This is due to the right angle between them, which makes each leg perpendicular to the other. Therefore, either leg can serve as the base or the height, simplifying the computation.

This formula highlights a critical geometry concept: the area is always one-half the product of the base and height in any triangle.

Legs of a Triangle

The two legs of a right triangle are the sides that join to form the right angle. In other types of triangles, the base and height determination might not be as straightforward, but in right triangles, they correspond to the legs.

• Each leg serves a dual role in being part of the right angle formation.
• These sides are crucial because they directly allow the calculation of area using the area formula.

The exercise specifically involves using these legs, given as 3.46 feet and 2.55 feet, to compute the area of the triangle. Remembering that in right triangles, the legs assume the role of base and height simultaneously makes calculating the area straightforward.

Mathematics Problem Solving

When solving mathematics problems, particularly involving geometry like triangles, it is essential to follow a systematic approach:

1. **Identify the Type of Triangle:** Recognizing the triangle as right allows the use of specific properties, notably the use of legs as base and height. 2. **Recall Relevant Formulas:** Have a good grasp of the formulas pertaining to what you're asked to find. Here, remembering the area formula for triangles is key. 3. **Substitute and Solve:** Insert the given values into the formula, methodically solve each step by step. This ensures accuracy in reaching the solution. 4. **Check Units and Results:** Always verify the units of measurement and ensure that the final result is logical and expressed in the proper units.

Following these steps makes tackling mathematical problems less daunting and increases your confidence in achieving the correct answer.