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 triangle is approximately \(1.317\,\text{m}^2\).

Step 1: Understanding the Problem

We need to find the area of an isosceles triangle with two equal sides of length \(0.986\,\text{m}\) and a base of \(0.884\,\text{m}\). We will use Heron's formula to calculate the area.

Step 2: Calculate the Semi-Perimeter

First, calculate the semi-perimeter \(s\) of the triangle: \[ s = \frac{a + b + b}{2} = \frac{0.884 + 0.986 + 0.986}{2} = 1.928\,\text{m} \]

Step 3: Apply Heron's Formula

Next, apply Heron's formula to find the area \(A\) of the triangle:\[ A = \sqrt{s(s-a)(s-b)(s-b)} \]Where \(a = 0.884\) and \(b = 0.986\).Substitute the values:\[ A = \sqrt{1.928(1.928 - 0.884)(1.928 - 0.986)(1.928 - 0.986)} \]

Step 4: Simplify the Expression

Simplify inside the square root:\[ A = \sqrt{1.928 \times 1.044 \times 0.942 \times 0.942} \]

Step 5: Calculate the Area

Finally, compute the product inside the square root:\[ A \approx \sqrt{1.928 \times 1.044 \times 0.942^2} = \sqrt{1.7357} \]\[ A \approx 1.317\,\text{m}^2 \]

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle that has two sides of equal length. These two sides are referred to as the "legs," while the third side is known as the "base." Understanding the properties of an isosceles triangle is important when calculating its area, as certain formulas and approaches specifically apply to this type of triangle.

Some characteristics of isosceles triangles are:

• Two angles opposite the equal sides are also equal.
• The altitude from the vertex angle bisects the base, creating two congruent right triangles within the isosceles triangle.
• Due to the equal sides, symmetry plays a crucial role in simplifying various calculations related to the triangle.

Recognizing these properties can significantly aid in visualizing and solving geometric problems involving isosceles triangles, like calculating areas or angles.

Area Calculation Using Heron's Formula

Heron's formula is a powerful tool for calculating the area of a triangle when you know the lengths of all three sides. This formula is especially helpful when the standard formulas, like base-height for a right or isosceles triangle, are not easily applicable.

The formula is as follows: \ A = \sqrt{s(s-a)(s-b)(s-c)} \, where \( s \) is the semi-perimeter and \( a, b, c \) are the sides of the triangle.

• Start by finding the semi-perimeter: add all three sides together and divide by 2.
• Substitute these values into Heron's formula to compute the area.
• The key advantage of Heron's formula is that it applies universally to all triangles, regardless of the type.

This approach ensures an accurate calculation of the area without needing heights or angles, making it flexible and widely usable.

Understanding the Semi-Perimeter

The semi-perimeter is an intermediate step often used in geometric formulas, including Heron's formula. It represents half of the triangle's perimeter and is calculated by summing all the side lengths and dividing the total by two.

Here's why the semi-perimeter is useful:

• By using the semi-perimeter, Heron's formula integrates all three sides naturally, enabling area calculations without additional measurements.
• It simplifies the number of operations needed since you only need a few calculations to apply Heron's formula.
• Understanding it helps in comprehensive geometry problems where multiple properties of triangles need evaluation.

Insight into using the semi-perimeter effectively enhances problem-solving capabilities in triangular geometry.