Problem 43 - Precalculus

Question

Find the area of each triangle with measures given. $$a=6, b=10, c=9$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 26.57 square units.

Step 1: Recall Heron's Formula

Heron's formula is used to find the area of a triangle when all three side lengths are known. The formula is \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]where \( s \) is the semi-perimeter of the triangle, calculated as \( s = \frac{a+b+c}{2} \).

Step 2: Calculate the Semi-Perimeter

First, find the semi-perimeter by adding all side lengths and dividing by two.\[ s = \frac{6 + 10 + 9}{2} = \frac{25}{2} = 12.5 \]

Step 3: Apply Heron's Formula

Substitute the side lengths and the semi-perimeter value into Heron's formula to calculate the area. \[ A = \sqrt{12.5(12.5-6)(12.5-10)(12.5-9)} \]\[ = \sqrt{12.5 \times 6.5 \times 2.5 \times 3.5} \]

Step 4: Simplify the Expression

Calculate the product inside the square root.\[ 12.5 \times 6.5 \times 2.5 \times 3.5 = 706.25 \]Now, take the square root of the product. \[ A = \sqrt{706.25} \]

Step 5: Calculate the Area

Finally, solve the square root to find the area of the triangle.\[ A \approx 26.57 \] square units.

Key Concepts

Semi-Perimeter

The semi-perimeter is a key part of Heron's formula. It's a term that might sound complex, but it's simply the sum of the lengths of all sides of a triangle divided by two. For any triangle with side lengths denoted as \( a \), \( b \), and \( c \), the semi-perimeter \( s \) is given by the formula:

• \( s = \frac{a+b+c}{2} \)

In the exercise, we have a triangle with sides \( a = 6 \), \( b = 10 \), and \( c = 9 \). By applying the semi-perimeter formula, we add these side lengths:

• \( a+b+c = 6 + 10 + 9 = 25 \)

Then, we divide by 2 to find the semi-perimeter:

• \( s = \frac{25}{2} = 12.5 \)

The semi-perimeter is particularly useful because it allows us to plug back into Heron's formula and helps simplify the process of finding the area of the triangle.

Triangle Area

Finding the area of a triangle when you know just the side lengths can seem like a tricky task. However, by using Heron's formula, we can achieve this without needing the height of the triangle. Heron's formula for calculating the triangle area \( A \) is:

• \( A = \sqrt{s(s-a)(s-b)(s-c)} \)

Here, \( s \) is the semi-perimeter which was calculated in the previous section. The process involves subtracting each of the triangle's sides from the semi-perimeter and then using these differences in the formula:

• \( s-a = 12.5 - 6 = 6.5 \)
• \( s-b = 12.5 - 10 = 2.5 \)
• \( s-c = 12.5 - 9 = 3.5 \)

Substituting these values, Heron's formula gives us:

• \( A = \sqrt{12.5 \times 6.5 \times 2.5 \times 3.5} \)

These calculations simplify the steps you'd otherwise need to find the height of the triangle, and directly lead to the area value.

Square Root

The square root is a mathematical tool crucial for calculating the triangle's area through Heron's formula. In simpler terms, taking the square root is like asking "what number squared (multiplied by itself) equals this number?" In our exercise, we need to find the square root of a calculated number inside Heron's formula. Let's break it down:

• The multiplication inside the square root gives us \( 706.25 \).

The next step involves taking the square root of this product:

• \( A = \sqrt{706.25} \)

This operation asks us "what number, when squared, equals 706.25?" After calculating, we find:

• \( A \approx 26.57 \)

Thus, the area of the triangle is approximately 26.57 square units. By understanding the square root process, you can easily transition from multiplication of numbers to finding precise results in geometric problems like these.