Problem 38 - Precalculus with Limits

Question

Using Heron's Area Formula use Heron's Area Formula to find the area of the triangle. $$ a=33, \quad b=36, \quad c=25 $$

Step-by-Step Solution

Answer

The area of the triangle is roughly 396 square units.

Step 1: Calculate the semi-perimeter

Add the sides of the triangle and divide by 2 to find the semi-perimeter. \(s = \frac{a+b+c}{2} = \frac{33+36+25}{2} = 47\)

Step 2: Apply Heron's Formula

Now apply the Heron's formula to find the area of the triangle. Here \(s = 47\), \(a = 33\), \(b = 36\), and \(c = 25\). \(A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{47(47-33)(47-36)(47-25)} = \sqrt{47*14*11*22}\)

Step 3: Calculate the Area

Now calculate the value under the radical sign. \(A = \sqrt{156764} \approx 396\)

Key Concepts

Semi-perimeter

The semi-perimeter of a triangle is a key component in Heron's Area Formula. It's the halfway point of the triangle's perimeter. Here's how it works:

• First, you add up the lengths of the three sides of the triangle.
• Then, divide that total by 2. This gives you the semi-perimeter, often represented by the variable \(s\).

For example, in our triangle with sides 33, 36, and 25, you first add these values together: \[33 + 36 + 25 = 94\]Next, divide by 2 to get the semi-perimeter:\[s = \frac{94}{2} = 47\]Understanding the semi-perimeter is crucial because it's used in the formula to calculate a triangle's area without knowing the height.

Triangle Geometry

Triangle geometry involves understanding the properties and relationships between a triangle's sides and angles. In the context of Heron's formula, the focus is on the lengths of the sides rather than the angles.

• The three sides of any triangle are typically denoted as \(a\), \(b\), and \(c\).
• The sum of any two sides must be greater than the third side. This is known as the triangle inequality theorem.

In our example, we checked the sides : \[a = 33, \quad b = 36, \quad c = 25\]First, we need to ensure they satisfy the triangle inequality. This means:

• \(33 + 36 > 25\)
• \(33 + 25 > 36\)
• \(36 + 25 > 33\)

Each condition is satisfied, so these are legitimate triangle side lengths. Understanding these fundamental properties helps ensure correct application of Heron's formula.

Radical Expression

In mathematical terms, a radical expression involves a root, such as square roots, cube roots, etc. Heron's formula includes a square root as it derives the area of a triangle from the semi-perimeter and its sides.Here's a breakdown of how the radical expression comes into play:

• Heron's formula itself is \(A = \sqrt{s(s-a)(s-b)(s-c)}\).
• In our example, this becomes \(\sqrt{47(47-33)(47-36)(47-25)}\).
• This means computing \(47 \times 14 \times 11 \times 22\), which equals \(156764\).

The last step is to find the square root of this large number, \(\sqrt{156764}\), which simplifies to approximately 396.Understanding how to manipulate these radical expressions is essential, as they appear in many areas of geometry beyond Heron's formula.