Problem 30 - Algebra and Trigonometry

Question

Find the area of the triangle whose sides have the given lengths. \(a=11, \quad b=100, \quad c=101\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 174 square units.

Step 1: Determine if a triangle is valid

To verify if the given sides can form a triangle, check the triangle inequality: the sum of the lengths of any two sides must be greater than the length of the third side. Here are the checks: 1. \(a + b > c\) which is \(11 + 100 > 101\) or \(111 > 101\), which is true. 2. \(a + c > b\) which is \(11 + 101 > 100\) or \(112 > 100\), which is true. 3. \(b + c > a\) which is \(100 + 101 > 11\) or \(201 > 11\), which is true. Thus, the sides can form a triangle.

Step 2: Calculate the semi-perimeter

The semi-perimeter \(s\) of a triangle with sides \(a\), \(b\), and \(c\) is calculated using the formula: \[ s = \frac{a + b + c}{2} \] Substitute the given values: \[ s = \frac{11 + 100 + 101}{2} = \frac{212}{2} = 106 \]

Step 3: Use Heron's Formula for area

Heron's formula for the area \(A\) of a triangle with sides \(a\), \(b\), and \(c\) and semi-perimeter \(s\) is: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] Substitute the known values: \[ A = \sqrt{106(106-11)(106-100)(106-101)} \] Simplify the expressions: \[ A = \sqrt{106 \times 95 \times 6 \times 5} \]

Step 4: Simplify and calculate

Calculate the expression inside the square root: \[ 106 \times 95 \times 6 \times 5 = 30270 \] Find the square root of the result to get the area: \[ A = \sqrt{30270} \approx 174 \] Thus, the area of the triangle is approximately 174 square units.

Key Concepts

Heron's formula

Heron's formula is a convenient way to find the area of a triangle when you know the lengths of all three sides and do not have the height readily available. It is especially handy for non-right triangles.
Before using Heron's formula, make sure the sides can form a valid triangle, which can be confirmed using the triangle inequality theorem.
This formula is:

• Compute the semi-perimeter, denoted as s.
• Use the formula: \[A = \sqrt{s(s-a)(s-b)(s-c)}\]

Here, A is the area of the triangle, and a, b, and c are the lengths of the sides.
In our example, substitute the values for a = 11, b = 100, and c = 101 to find the area using the semi-perimeter s calculated earlier.

triangle inequality theorem

Before attempting to calculate the area with Heron's formula, it's crucial to check if the sides can indeed form a triangle. This is where the triangle inequality theorem comes into play. It helps determine if any three given lengths can be sides of a triangle.
The principle states:

• The sum of the lengths of any two sides must be greater than the length of the remaining side.
• You need to check all three combinations:
  • \(a + b > c\)
  • \(a + c > b\)
  • \(b + c > a\)

Once all these conditions are met, as in our problem where the sides are 11, 100, and 101, we can proceed to compute other properties, such as area, since these values accurately form a triangle.

semi-perimeter calculation

Once you've established that a valid triangle can be formed, the next step in using Heron's formula is calculating the semi-perimeter of the triangle.
The semi-perimeter is essentially half of the triangle's perimeter and is crucial in calculating area with Heron's formula.
Calculate it using the formula:

• \[s = \frac{a + b + c}{2}\]
• Where a, b, c are the lengths of the sides.

For the triangle with sides 11, 100, and 101, the semi-perimeter is calculated as follows:

• Add the side lengths: \(11 + 100 + 101 = 212\)
• Divide by 2:\(s = \frac{212}{2} = 106\)

This calculated semi-perimeter, 106 in this case, then feeds into Heron's formula to determine the area. It is an essential step that helps simplify the subsequent calculations.