Problem 45 - Algebra 2 and Trigonometry

Question

The lengths of the legs of a right triangle in feet are \((3+\sqrt{3})\) and \((3-\sqrt{3})\) a. Find the length of the hypotenuse of the triangle. b. Express, in simplest form, the perimeter of the triangle. c. What is the area of the triangle?

Step-by-Step Solution

Answer

Hypotenuse: \(2\sqrt{6}\); Perimeter: \(6 + 2\sqrt{6}\); Area: 3 square feet.

Step 1: Identify the sides of the triangle

The problem provides us with the leg lengths of a right triangle: \((3+\sqrt{3})\) and \((3-\sqrt{3})\). We'll use these to find the hypotenuse.

Step 2: Use the Pythagorean Theorem to find the hypotenuse

The Pythagorean Theorem states that in a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. Thus, \(c = \sqrt{(3+\sqrt{3})^2 + (3-\sqrt{3})^2}\).

Step 3: Calculate each squared term separately

Calculate \((3+\sqrt{3})^2 = 9 + 6\sqrt{3} + 3\) and \((3-\sqrt{3})^2 = 9 - 6\sqrt{3} + 3\). After simplification, each term equals 12.

Step 4: Sum the squared terms

Add the squared terms as follows: \(12 + 12 = 24\).

Step 5: Find the square root to get the hypotenuse

So, the hypotenuse \(c = \sqrt{24}\). Simplify this to \(2\sqrt{6}\) for the hypotenuse length.

Step 6: Calculate the perimeter of the triangle

The perimeter of the triangle is the sum of all sides: \((3+\sqrt{3}) + (3-\sqrt{3}) + 2\sqrt{6}\). Here, the simplified sum of the first two terms is 6, leading to the expression \(6 + 2\sqrt{6}\).

Step 7: Calculate the area of the triangle

The area of a right triangle is given by \(\text{Area} = \frac{1}{2} \times a \times b\). Here, \(a = 3+\sqrt{3}\) and \(b = 3-\sqrt{3}\). Thus, calculate \(\text{Area} = \frac{1}{2} \times (3+\sqrt{3}) \times (3-\sqrt{3})\).

Step 8: Simplify the area expression

The expression \((3+\sqrt{3}) \times (3-\sqrt{3})\) simplifies to \(9 - 3\), which is 6. Therefore, the area is \(\frac{1}{2} \times 6 = 3\).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This unique angle is what sets it apart from other triangles. In a right triangle, the side opposite this angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides are referred to as the legs.

These legs are crucial in applying the Pythagorean Theorem, which is the cornerstone for solving many problems involving right triangles. The theorem helps us determine the relationships between the side lengths of the triangle.

Remember, knowing just two side lengths of a right triangle allows you to find the third side using this beautiful theorem!

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. To find the hypotenuse, we use the Pythagorean Theorem:

• According to the theorem: \( a^2 + b^2 = c^2 \), where \(c\) represents the hypotenuse.
• In our example, the legs are \((3+\sqrt{3})\) and \((3-\sqrt{3})\).
• So, \(c = \sqrt{(3+\sqrt{3})^2 + (3-\sqrt{3})^2} = \sqrt{24} = 2\sqrt{6}\).

Once you have the lengths of these legs, using the theorem helps you find the hypotenuse with ease.

Triangle Perimeter

The perimeter of a triangle is the total length around it, found by summing up all its sides. For our right triangle:

• You have the two leg lengths \((3+\sqrt{3})\) and \((3-\sqrt{3})\).
• After finding the hypotenuse \(2\sqrt{6}\), simply add these together:
• \( (3+\sqrt{3}) + (3-\sqrt{3}) + 2\sqrt{6} = 6 + 2\sqrt{6} \).

By simplifying terms when possible, calculating perimeter becomes straightforward. This step is helpful when you need to set up boundaries or solve problems requiring the total exterior distance of the triangle.

Triangle Area

The area of a triangle gives you the size of the space it encloses. For a right triangle, the area is easily calculated using the formula:

• The area \(= \frac{1}{2} \times \text{base} \times \text{height} \).
• Here, the base and height are the two legs of the triangle, which are \((3+\sqrt{3})\) and \((3-\sqrt{3})\).
• Multiplying these: \((3+\sqrt{3}) \times (3-\sqrt{3}) = 9 - 3 = 6 \).
• Insert into the area formula: \(\frac{1}{2} \times 6 = 3 \).

This numerical value provides insight into the space covered by the triangle, and understanding it can be quite useful in various practical applications where space calculation is necessary.