Problem 48 - Precalculus

Question

Find the area of each triangle with measures given. $$a=\sqrt{2}, b=\sqrt{3}, c=\sqrt{5}$$

Step-by-Step Solution

Answer

Use Heron's formula with given side lengths to find the area.

Step 1: Understand the Given Dimensions

We have a triangle with side lengths: \(a = \sqrt{2}\), \(b = \sqrt{3}\), and \(c = \sqrt{5}\). We will use these side lengths to find the area of the triangle using Heron's formula.

Step 2: Calculate the Semi-Perimeter

The semi-perimeter \(s\) of the triangle is calculated as \(s = \frac{a+b+c}{2}\). Substituting the given side lengths, we have:\[s = \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{2}\]

Step 3: Apply Heron's Formula

Heron's formula for the area \(A\) of a triangle with sides \(a\), \(b\), and \(c\) is given by \[A = \sqrt{s(s-a)(s-b)(s-c)}\]Substitute the semi-perimeter and side lengths:\[A = \sqrt{\left( \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{2} \right) \left( \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{2} - \sqrt{2} \right) \left( \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{2} - \sqrt{3} \right) \left( \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{2} - \sqrt{5} \right)}\]

Step 4: Simplify and Calculate

Calculate each term inside the square root:- \(s - a\) becomes \(\frac{\sqrt{3} + \sqrt{5} - \sqrt{2}}{2}\)- \(s - b\) becomes \(\frac{\sqrt{2} + \sqrt{5} - \sqrt{3}}{2}\)- \(s - c\) becomes \(\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{2}\)Multiply these together and solve the expression under the square root. This step will require detailed calculation and simplification to arrive at the final area. This step requires algebraic manipulation skills to simplify the product and take the square root.

Step 5: Present the Final Area

After calculating and simplifying the expression under the square root, determine the value of the area \(A\) of the triangle accurately. Adopting numerical methods or approximation if necessary, due to the complexity of the calculations. For an accurate result, evaluate using a calculator if required.

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle can be straightforward when using Heron's formula. Heron's formula is especially useful when all three side lengths of a triangle are known, but the height is not. This method utilizes the semi-perimeter and the side lengths to determine the area.
First, gather the side lengths of the triangle. For example, sides could be known values like \(a = \sqrt{2}\), \(b = \sqrt{3}\), and \(c = \sqrt{5}\). Next, calculate the semi-perimeter and substitute it and the side lengths into Heron's formula: \[A = \sqrt{s(s-a)(s-b)(s-c)}\]where \(s\) is the semi-perimeter found using \(s = \frac{a+b+c}{2}\).
This process involves substituting the known values, performing calculations inside the square root first, and then solving for the area. Simplifying the algebraic expressions accurately is crucial for obtaining the correct area.

Semi-Perimeter

The semi-perimeter of a triangle is a crucial concept in calculating its area using Heron's formula. It is simply half of the triangle's perimeter. In other words, it's the sum of the lengths of the sides of the triangle divided by two.
This is particularly useful because it provides a way to further break down the computation of the area into more manageable parts.

• To find the semi-perimeter \(s\), use the formula: \[s = \frac{a + b + c}{2}\]
• Where \(a\), \(b\), and \(c\) are the side lengths of the triangle.

For example, with side lengths \(a = \sqrt{2}\), \(b = \sqrt{3}\), and \(c = \sqrt{5}\), the calculation would be:\[s = \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{2}\] This semi-perimeter is then used directly in Heron's formula to calculate the area. It acts as a bridge to integrate the known dimensions of the triangle into the area calculation process.

Square Roots in Algebra

Square roots are an essential part of algebra, especially when dealing with geometric shapes like triangles. In this context, square roots help in handling various conditions where values are not expressed as integers or simple fractions.
For instance, the side lengths of a triangle can be square roots, like \(a = \sqrt{2}\), \(b = \sqrt{3}\), and \(c = \sqrt{5}\). These values need careful management when performing operations.

• Simplify expressions under square root signs where possible to make calculations easier.
• Be accurate when adding, subtracting, or multiplying square roots, as these operations need consideration of like terms.

In Heron's formula, once you find the semi-perimeter and the necessary subtractions (\(s-a\), \(s-b\), and \(s-c\)), you will multiply them and take the square root of their product. This includes collecting like radicals and ensuring the terms inside the square root are simplified properly. This is where knowing how to manipulate square roots and handle algebraic expressions becomes an essential skill for solving the problem accurately.