91Ó°ÊÓ

The triangle inequality theorem is a foundational concept in geometry that helps determine whether three side lengths can construct a triangle. According to the theorem, for a set of three sides to form a triangle, the sum of the lengths of any two sides must exceed the length of the third side. This rule guarantees that a valid and closed figure is achieved.
To explore this further, consider a triangle with sides labeled as \(a\), \(b\), and \(c\). The theorem can be tested using these inequalities:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

If each condition is satisfied, the sides can form a triangle. For instance, using \(a = 11\), \(b = 100\), and \(c = 101\) in these inequalities shows they satisfy all requirements, confirming you have a valid triangle.

The semi-perimeter of a triangle, represented by the symbol \(s\), is half of the triangle's perimeter. Consider it as a step that simplifies calculations when using formulas like Heron's formula to find the triangle's area.
To calculate the semi-perimeter of a triangle with sides \(a\), \(b\), and \(c\), use the formula:

• \[ s = \frac{a+b+c}{2} \]

This provides a single value representing average side length-related information, streamlining further computations.
In the given problem, for a triangle with sides \(a = 11\), \(b = 100\), and \(c = 101\), the semi-perimeter is found as follows:

• \[ s = \frac{11 + 100 + 101}{2} = 106 \]

Now equipped with the semi-perimeter, you can move to the next step in determining the triangle's area using Heron's formula.

Calculating the area of a triangle can be straightforward, especially with Heron's formula, which relies on the side lengths and semi-perimeter. This method is effective when height and base are not easily determined. Begin by confirming the triangle's side lengths and calculate the semi-perimeter, \(s\).
Apply Heron's formula, which is as follows:

• \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

This formula allows you to compute the area by substituting known values. For the triangle with side lengths \(a = 11\), \(b = 100\), and \(c = 101\) and semi-perimeter \(s = 106\), solve the following:

• Calculate \(s - a\), \(s - b\), \(s - c\): \[ s-a = 106-11, \ s-b = 106-100, \ s-c = 106-101 \]
• Then, find \( (s-a) \times (s-b) \times (s-c) \)
• Finally, insert these into the formula to yield: \[ A = \sqrt{106 \times 95 \times 6 \times 5} = 550 \text{ square units} \]

Thus, Heron's formula gives the triangle's area effectively, even when traditional methods aren't applied.