Problem 31 - Algebra and Trigonometry

Question

Find the area of the triangle whose sides have the given lengths. \(a=7, \quad b=8, \quad c=9\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 26.83 square units.

Step 1: Understand Heron's Formula

To find the area of a triangle when the lengths of all three sides are known, we use Heron's formula. Heron's formula states that the area of a triangle with sides of length \(a\), \(b\), and \(c\) is given by \(A = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s\) is the semi-perimeter of the triangle.

Step 2: Calculate the Semi-perimeter

First, calculate the semi-perimeter \(s\) of the triangle. This is half the sum of the side lengths:\[s = \frac{a + b + c}{2}\]Substituting in the given values:\[s = \frac{7 + 8 + 9}{2} = 12.\]

Step 3: Substitute Values into Heron's Formula

Now that we have the semi-perimeter, substitute \(s\), \(a\), \(b\), and \(c\) into Heron's formula:\[A = \sqrt{12(12-7)(12-8)(12-9)}\]

Step 4: Simplify the Expression

Simplify the expression under the square root:\[A = \sqrt{12 \times 5 \times 4 \times 3}\]Calculate this step-by-step:- \(12 \times 5 = 60\)- \(60 \times 4 = 240\)- \(240 \times 3 = 720\)So, \(A = \sqrt{720}\).

Step 5: Calculate the Square Root

Finally, calculate the square root of 720 to find the area of the triangle:\[\sqrt{720} \approx 26.83\]Thus, the area of the triangle is approximately 26.83 square units.

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle using Heron's Formula is a useful technique when you know all three sides of the triangle. Heron's Formula helps find the area without needing to measure altitudes or angles. In the given exercise, the sides of the triangle are 7, 8, and 9 units long. Start by finding the semi-perimeter, then apply Heron's Formula to determine the area.

• Calculate the semi-perimeter: \[ s = \frac{a+b+c}{2} \] where \( a = 7 \), \( b = 8 \), \( c = 9 \).
• Using Heron's Formula after finding the semi-perimeter lets you calculate the area \( A = \sqrt{s(s-a)(s-b)(s-c)} \).

By substituting these values into the formula, you simplify it to find the area. This method is efficient for solving triangles that are not right-angled and don't have conveniently measurable heights.

Semi-perimeter

The semi-perimeter is a critical step in using Heron's Formula. It simplifies the problem of finding the area of a triangle when you only know its sides. Essentially, the semi-perimeter is half the perimeter of the triangle. You calculate it by adding up all three sides, then dividing by two.

• For our triangle, with sides 7, 8, and 9, the semi-perimeter \( s \) is calculated as: \[ s = \frac{7 + 8 + 9}{2} = 12 \]
• Understanding the semi-perimeter helps because it makes the subsequent use of Heron's Formula straightforward.

Once you have the semi-perimeter, it's easier to substitute values into the formula and solve for the area. Calculating the semi-perimeter allows us to transform the formula into something manageable and keeps complex numbers at bay.

Basic Algebra

Algebra plays a key role in simplifying and solving the expressions found in Heron's Formula. Once you have calculated the semi-perimeter \( s \), the next step involves substituting values into the formula and simplifying the expression.

• From the semi-perimeter, substitute into the formula: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]After you've inserted the numbers (e.g., \( s = 12 \), \( s-a = 5 \), etc.), it's crucial to simplify the arithmetic operations inside the square root.
• Simplification skills are essential, so ensure you multiply correctly in steps, \( 12 \times 5 \times 4 \times 3 \), before taking the square root.

Mastering basic algebra is necessary for executing these calculations seamlessly, which leads to determining the triangle's area accurately. This process highlights your understanding of algebraic manipulation in geometric contexts.