Problem 46 - Precalculus

Question

Find the area of each triangle with measures given. $$a=146.5, b=146.5, c=100$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 7151.96 square units.

Step 1: Recognize the triangle type

The given side lengths are \(a = 146.5\), \(b = 146.5\), and \(c = 100\). Since two sides are equal \(a = b\), it is an isosceles triangle.

Step 2: Use Heron's Formula

To find the area of a triangle when the side lengths are known, Heron's formula is a great choice. First, calculate the semi-perimeter \(s\): \[ s = \frac{a+b+c}{2} = \frac{146.5 + 146.5 + 100}{2} = 196.5 \]

Step 3: Apply Heron's formula

With \(s = 196.5\), plug the values into Heron's formula: \[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \] Calculate each term:- \(s-a = 196.5 - 146.5 = 50\)- \(s-b = 196.5 - 146.5 = 50\)- \(s-c = 196.5 - 100 = 96.5\)Now, compute the area:\[ \text{Area} = \sqrt{196.5(50)(50)(96.5)} \]

Step 4: Simplify and calculate the result

Calculate the square root:\[ \text{Area} = \sqrt{196.5 \times 50 \times 50 \times 96.5} \]Computing the values, the approximate area is:\[ \text{Area} \approx 7151.96 \]

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle that has at least two sides of equal length. In the example provided, the side lengths are given as: \( a = 146.5 \), \( b = 146.5 \), and \( c = 100 \). Here, two sides \( a \) and \( b \) are equal, thus confirming it is an isosceles triangle. This characteristic leads to certain properties, such as:

• The angles opposite the equal sides are also equal.
• An axis of symmetry exists along the median from the vertex angle to the base.

Understanding the definition of an isosceles triangle helps in making use of specific formulas and insights while solving problems related to triangles, as some calculations are simplified when dealing with two equal sides.`
Recognizing the triangle type is crucial before applying any further calculations, as it can dictate the method and equations used.`

Semi-perimeter

The concept of the semi-perimeter is vital when working with Heron's formula, a formula used to calculate the area of a triangle when all side lengths are known. The semi-perimeter \( s \) is half the total length of the sides of the triangle. It is calculated as follows:`
\[ s = \frac{a+b+c}{2} \] This step is crucial as it sets the stage for Heron's formula by simplifying the calculations for determining a triangle's area. In the given triangle exercise, this was calculated as: \[ s = \frac{146.5 + 146.5 + 100}{2} = 196.5\] Knowing the semi-perimeter helps in efficiently applying Heron's formula and drastically reduces the complexity associated with computing the triangle's area.`
Heron's formula cannot proceed without this important intermediate step, making the concept of the semi-perimeter integral to solving such problems.

Area of a Triangle

The area of a triangle can be calculated in many ways based on the information available about the triangle. For triangles where side lengths are known, Heron's formula is particularly useful. `This formula uses all three side lengths and the semi-perimeter to find the area. `The general form of Heron's formula is: `
\[ ext{Area} = \sqrt{s(s-a)(s-b)(s-c)} \] Where \( s \) is the semi-perimeter. `In the provided exercise, this calculation was carried out using these steps:`

• Compute \( s-a \), \( s-b \), and \( s-c \).
• Substitute these values into the square root expression.
• Simplify to find the computed area.

The specific calculations give us:\[ ext{Area} = \sqrt{196.5 \times 50 \times 50 \times 96.5} \] Resulting in an approximate area of \( 7151.96 \). `By systematically applying Heron's formula, we were able to determine the area accurately. This method is valuable as it accommodates any triangle, unlike other formulas that may require specific height information.`