Problem 15 - Basic Technical Mathematics with Calculus

Question

Find the area of each triangle. Isosceles triangle, equal sides of \(0.986 \mathrm{m}\), third side of \(0.884 \mathrm{m}\)

Step-by-Step Solution

Answer

The area of the isosceles triangle is approximately 0.3905 square meters.

Step 1: Identify the Type of Triangle

This problem involves an isosceles triangle, which means two of its sides are of equal length. Given the side lengths, the equal sides are each 0.986 meters, and the base is 0.884 meters.

Step 2: Use Heron's Formula

To find the area of the triangle, we will use Heron's formula. First, calculate the semi-perimeter of the triangle by adding up all sides and dividing by 2. \[s = \frac{0.986 + 0.986 + 0.884}{2}\]Evaluate this to find: \[s = 1.428 \, \text{meters}\]

Step 3: Calculate the Area Using Heron's Formula

Plug in the semi-perimeter (\(\text{s}=1.428\)) and the side lengths into Heron's formula: \[A = \sqrt{s(s-a)(s-b)(s-c)}\]Here, \(a = 0.986\), \(b = 0.986\), and \(c = 0.884\). Thus, \[A = \sqrt{1.428(1.428 - 0.986)(1.428 - 0.986)(1.428 - 0.884)}\]Calculate each part: \[A = \sqrt{1.428(0.442)(0.442)(0.544)}\]\[A = \sqrt{0.1524}\]

Step 4: Simplify the Expression

Calculate the square root of 0.1524 to find the area of the triangle:\[A \approx 0.3905 \, \text{square meters} \]

Key Concepts

Isosceles Triangle

An isosceles triangle is a special kind of triangle where two sides are of equal length. These sides are often referred to as the 'legs' of the triangle, while the third side is known as the 'base.' Understanding this property helps us identify and solve many problems in geometry, including calculating areas and angles.
In this particular triangle, both the legs measure 0.986 meters, and the base measures 0.884 meters. Knowing it's isosceles gives us specific shortcuts in calculations and predictions about angles, which can be useful in real-world situations like engineering and architecture.

Heron's Formula

Heron's formula is a handy tool for finding the area of a triangle when we know the lengths of all three sides. It's especially useful when the triangle isn't a right triangle, making other methods less straightforward.
Heron's formula states that the area of a triangle with sides of length a, b, and c can be calculated using the semi-perimeter \(s\):

• First, find the semi-perimeter: \(s = \frac{a+b+c}{2}\)
• Then, use the formula: \(A = \sqrt{s(s-a)(s-b)(s-c)}\)

This method provides flexibility and simplicity in obtaining the area of complex triangles, making it an essential concept in geometry.

Semi-Perimeter

The semi-perimeter of a triangle is half of its perimeter. It's particularly vital when using Heron's formula, as it serves as the foundation for further calculations.
To find it, simply add up all the side lengths of the triangle and divide the result by two. For example, with sides of 0.986 meters, 0.986 meters, and 0.884 meters, the semi-perimeter \(s\) is calculated as follows: \[s = \frac{0.986 + 0.986 + 0.884}{2} = 1.428 \] meters.
This value simplifies the computation of areas and is a stepping stone to employing Heron's formula effectively.

Area Calculation

Using Heron's formula, area calculation becomes efficient and straightforward, even for non-right triangles. Once the semi-perimeter is established, the real magic happens as we find the actual area.
Insert the semi-perimeter and each side's length into Heron's formula: \[A = \sqrt{s(s-a)(s-b)(s-c)}\] where \(a = 0.986\), \(b = 0.986\), and \(c = 0.884\).
Calculate: \[A = \sqrt{1.428(1.428 - 0.986)(1.428 - 0.986)(1.428 - 0.884)}\] resulting in: \[A = \sqrt{0.1524} \approx 0.3905 \, \text{square meters}\]
This method shows how all the components—semi-perimeter, specific calculations, and understanding of isosceles properties—come together to easily determine the area.