Problem 5 - Algebra and Trigonometry

Question

Find the area of the right triangle whose legs are of length 3 and 4

Step-by-Step Solution

Answer

The area of the triangle is 6 square units.

Step 1 - Understand the formula for the area of a right triangle

The area of a right triangle can be found using the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). In this case, the legs of the triangle represent the base and height.

Step 2 - Identify the base and height of the triangle

For this triangle, the legs are given as 3 and 4. We can assign 3 as the base and 4 as the height, or vice versa, as the order does not matter for multiplication.

Step 3 - Substitute the values into the formula

Using the formula for the area: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Substitute the base = 3 and height = 4 into the equation: \( \text{Area} = \frac{1}{2} \times 3 \times 4 \).

Step 4 - Perform the calculation

Now, multiply the values: \( \text{Area} = \frac{1}{2} \times 3 \times 4 \) \( \text{Area} = \frac{1}{2} \times 12 \) \( \text{Area} = 6 \).

Key Concepts

right triangle

A right triangle is a type of triangle that has one of its angles exactly equal to 90 degrees. This right angle makes many calculations simpler and is a fundamental concept in geometry. The two sides forming the right angle are known as the 'legs' of the triangle, and these are key to various geometric formulas and calculations. The longest side, opposite the right angle, is known as the hypotenuse. The defining characteristic of a right triangle makes it a cornerstone for trigonometric functions and various calculations in different fields of science and engineering.

area calculation

Calculating the area of a right triangle is straightforward. The general formula to find the area of any triangle is \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). For a right triangle, the base and the height are the two legs. Here’s how to calculate it step-by-step:

• Identify the legs of the triangle.
• One leg can serve as the base, and the other as the height.
• Use the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).

This formula works due to the right angle, simplifying what would otherwise be a more complex calculation for other types of triangles.

geometric formulas

Geometric formulas are crucial in determining various properties of shapes and figures. For right triangles, the key formula to remember for area calculation is \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). This formula is derived from more general principles of geometry but is particularly easy to use with right triangles. Knowing this, you can quickly find the area as long as the measurements of the legs are provided. This method relies on the fundamental principle that any triangle’s area is half the product of its base and height, and it holds true across different applications, whether in construction, design, or theoretical problems.

substitution in formulas

Substitution in formulas is an essential mathematical skill. It involves replacing variables in a formula with their actual numerical values. When using the area formula for a right triangle: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\), you perform the following steps:

• Assign the given measurements to the base and height. In our example, let’s say the base is 3 and the height is 4.
• Substitute these values into the formula: \( \text{Area} = \frac{1}{2} \times 3 \times 4 \).
• Carry out the multiplication: \( \text{Area} = \frac{1}{2} \times 12 \).
• Simplify the result if needed, to get the final area: \( \text{Area} = 6 \).

This straightforward process of substitution simplifies working with abstract formulas, making tangible results accessible and understandable.