Problem 30 - Algebra and Trigonometry

Question

Find the area of the triangle whose sides have the given lengths. \(a=1, \quad b=2, \quad c=2\)

Step-by-Step Solution

Answer

The area of the triangle is \( \frac{\sqrt{15}}{4} \).

Step 1: Verify if the Triangle is Possible

First, we need to check if a triangle with sides 1, 2, and 2 is possible. According to the triangle inequality theorem, the sum of two sides must be greater than the third side. Check the conditions: \(1 + 2 > 2\), \(1 + 2 > 2\), \(2 + 2 > 1\). All conditions are satisfied, so the triangle is possible.

Step 2: Identify the Type of Triangle

The given side lengths are 1, 2, and 2. Since two sides are equal, this is an isosceles triangle. Identifying the type of triangle helps simplify calculations as specific formulas can be used for certain triangle types.

Step 3: Use Heron's Formula for Area

Since we confirmed that the triangle is possible, we will use Heron's formula. First, calculate the semi-perimeter \(s\) using \(s = \frac{a+b+c}{2}\). Substitute the side lengths: \(s = \frac{1+2+2}{2} = \frac{5}{2}\).

Step 4: Calculate the Area

Now, apply Heron's formula: \( A = \sqrt{s(s-a)(s-b)(s-c)} \). Substitute the known values: \(A = \sqrt{\frac{5}{2}\left(\frac{5}{2}-1\right)\left(\frac{5}{2}-2\right)\left(\frac{5}{2}-2\right)}\). Simplifying, we get \(A = \sqrt{\frac{5}{2}\cdot\frac{3}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}}\). After calculating, the area \(A = \sqrt{\frac{15}{16}}\).

Step 5: Finalize the Calculation

Compute the simplified form: \( A = \frac{\sqrt{15}}{4} \). This is the final area of the triangle with sides 1, 2, and 2.

Key Concepts

Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental rule in geometry that determines whether three given lengths can form a triangle. For any three sides to constitute a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This ensures the sides can connect to enclose a space and form a valid triangle.

For example, if you have sides of lengths 1, 2, and 2, it's important to check the following conditions:

• 1 + 2 > 2
• 1 + 2 > 2
• 2 + 2 > 1

In this case, all conditions are satisfied, meaning a triangle with these side lengths is indeed possible. This theorem helps avoid attempting to solve problems involving unrealistic triangle dimensions.

Isosceles Triangle

An isosceles triangle is a triangle with at least two sides of equal length. This type of triangle is known for its symmetry, which can simplify certain geometric calculations. In the context of our example, with side lengths of 1, 2, and 2, notice that two sides are equal, forming an isosceles triangle.

Properties of an isosceles triangle include:

• Two equal sides, known as the legs.
• An equal base angle opposite each leg.
• A different third side called the base.

Recognizing that a triangle is isosceles can be useful. It allows for the use of specific formulas and theorems tailored for these triangles, such as methods to find angles or area that involve fewer calculations.

Heron's Formula

Heron’s Formula is a mathematical formula used to calculate the area of a triangle when all three sides are known. This makes it particularly handy for cases where the height of the triangle is not easily determined.

Here’s how you use Heron's Formula:

• First, calculate the semi-perimeter (\( s \)). This is \( s = \frac{a + b + c}{2} \).
• Then, find the area \( A \) using \( A = \sqrt{s(s-a)(s-b)(s-c)} \).

In the problem described, the sides 1, 2, and 2 lead to a semi-perimeter of \( \frac{5}{2} \). Applying Heron's Formula, the computation becomes straightforward. This method is universally applicable to any shape or size of triangle, reinforcing its versatility and utility.

Semi-perimeter

The semi-perimeter of a triangle is half the sum of its side lengths. It's often denoted by the symbol \( s \) and is a key component in various geometric calculations, including Heron's Formula.

To find the semi-perimeter, you simply sum the lengths of all sides and divide by two. For the triangle with sides 1, 2, and 2, the semi-perimeter is calculated as follows: \( s = \frac{1 + 2 + 2}{2} = \frac{5}{2} \).

Using the semi-perimeter:

• Makes it easier to apply Heron’s Formula.
• Compresses the complexity of handling triangle dimensions into a simpler number.

The concept of semi-perimeter streamlines the process of using formulas that might otherwise be cumbersome, allowing for efficient problem-solving.