Problem 10 - Precalculus: Functions and Graphs

Question

Approximate the area of triangle \(A B C\) to the nearest 0.1 square unit. $$a=4, \quad b=7, \quad c=10$$

Step-by-Step Solution

Answer

The area of triangle ABC is approximately 10.9 square units.

Step 1: Understand Heron's Formula

To find the area of a triangle when the lengths of all three sides are known (denoted as \(a\), \(b\), and \(c\)), we use Heron's formula. This formula states that the area \(A\) is given by: \[A = \sqrt{s(s-a)(s-b)(s-c)}\]where \(s\) is the semi-perimeter of the triangle, calculated as \(s = \frac{a+b+c}{2}\).

Step 2: Calculate the Semi-Perimeter

Given the side lengths \(a = 4\), \(b = 7\), and \(c = 10\), we first find the semi-perimeter \(s\):\[s = \frac{4+7+10}{2} = \frac{21}{2} = 10.5\]

Step 3: Apply Heron's Formula

Now that we have the semi-perimeter, we use it in Heron's formula to find the area:\[A = \sqrt{10.5\times(10.5 - 4)\times(10.5 - 7)\times(10.5 - 10)} = \sqrt{10.5\times 6.5\times 3.5\times 0.5}\]

Step 4: Simplify the Calculation

Calculate the expression inside the square root first:\[10.5 \times 6.5 \times 3.5 \times 0.5 = 119.4375\]

Step 5: Find the Area Approximation

Finally, compute the square root of the result from the previous step:\[A = \sqrt{119.4375} \approx 10.928\]Rounding to the nearest 0.1, the area is approximately 10.9 square units.

Key Concepts

Triangle Area Calculation

When you have a triangle and you know the lengths of its sides, you can use Heron's Formula to find its area. Heron's formula is a handy tool because it doesn't require you to find angles or heights. The formula is: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]This formula helps you calculate the area using the semi-perimeter and the lengths of the sides. It fits perfectly when you know all three sides of the triangle. Let's see how it works in steps:

• First, find the semi-perimeter \(s\)
• Next, subtract each side from the semi-perimeter \(s\)
• Multiply these values and take the square root to get the area

Let's break it down further with the semi-perimeter in the next section.

Semi-Perimeter

The semi-perimeter \(s\) is helpful when using Heron's Formula. It simplifies the calculation by giving you a single number to work with. To calculate the semi-perimeter, take the sum of the three side lengths and divide by two: \[ s = \frac{a+b+c}{2} \]In our example, with sides \(a = 4\), \(b = 7\), and \(c = 10\), the semi-perimeter is:\[ s = \frac{4+7+10}{2} = 10.5 \]This value represents half the perimeter of the triangle, which is used to set up the next part of Heron's formula. It's important because it essentially becomes part of each term inside the square root when calculating the area. By understanding this step clearly, you prepare yourself to find the area with ease.

Approximation of Square Roots

After computing the expression inside Heron's formula, which often results in a complex number, the next step is to take the square root. Many times, especially in exercises, square roots don't simplify neatly. That's where approximation comes in.Once the semi-perimeter \(s\) and the side differences \((s-a), (s-b), (s-c)\) are calculated, they multiply to a specific product. In our example, we ended up with:\[ 10.5 \times 6.5 \times 3.5 \times 0.5 = 119.4375 \]To find the area, we take the square root and approximate:\[ A = \sqrt{119.4375} \approx 10.928 \]By rounding this to the nearest tenth, we get the area as approximately 10.9 square units. Using approximation helps us express the area in a more understandable way when the numbers don't work out perfectly.