Problem 26 - Algebra and Trigonometry

Question

Use Heron's formula to find the area of each triangle. Round to the nearest square unit. \(a=5\) feet \(, b=5\) feet, \(c=4\) feet

Step-by-Step Solution

Answer

The area of the triangle is approximately 9 square feet.

Step 1: Calculation of Semi-Perimeter (s)

The sum of the lengths of the sides of the triangle is 5+5+4 which is 14 feet. The semi-perimeter (s) is half of the total perimeter. So s = (14 feet)/2 = 7 feet.

Step 2: Substitution into Heron's formula

Now, substitute the values of a, b, c and s into Heron's formula to compute the area. The area formula becomes : Area = sqrt[7(7-5)(7-5)(7-4)]

Step 3: Simplify the expression

Simplify the above expression inside the square root to get: Area = sqrt[7(2)(2)(3)]

Step 4: Calculate the Area

Finally, multiply out and calculate the area: Area = sqrt[84] which approximately equals to 9.17 square feet (rounded to the nearest square unit).

Key Concepts

Understanding Triangles

Triangles are one of the basic shapes in geometry. A triangle comprises three sides and three angles. Common types of triangles include:

• Equilateral triangle: all sides and angles are equal.
• Isosceles triangle: two sides are equal, and two angles are equal.
• Scalene triangle: all sides and angles are different.

Every triangle also has a perimeter, which is simply the sum of the lengths of all its sides. Understanding these basics helps in calculating different properties like area. For example, in our exercise, the given triangle has sides measuring 5 feet, 5 feet, and 4 feet, making it an isosceles triangle.

Semi-Perimeter Concept

The semi-perimeter is a crucial concept when using Heron's formula. It is half of the perimeter of a triangle. Calculate it by adding up all the sides and dividing by two. For a triangle with side lengths of 5 feet, 5 feet, and 4 feet, the perimeter would be 14 feet. To find the semi-perimeter, divide this sum by two: \[ s = \frac{14}{2} = 7 \text{ feet} \] Understanding the semi-perimeter is important because it is a key component in Heron's formula, which allows us to find the area without needing to know the height or angles of the triangle.

Area Calculation Using Heron's Formula

Heron's formula is a practical tool for finding the area of a triangle when you know all three sides. The formula is: \[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \] Where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the sides of the triangle. Let's use this formula to calculate the area of the triangle from the exercise, where \( s = 7 \), \( a = 5 \), \( b = 5 \), and \( c = 4 \). Substituting these values into Heron's formula gives: \[ \text{Area} = \sqrt{7(7-5)(7-5)(7-4)} \] You then simplify inside the square root: \[ \text{Area} = \sqrt{7 \times 2 \times 2 \times 3} = \sqrt{84} \] Finally, calculating the square root provides an area of approximately 9.17 square feet, rounded to 9 square feet for simplicity. This method is valuable when direct measurement of the height is not feasible.