Problem 28 - Algebra and Trigonometry

Question

Find the area of the triangle whose sides have the given lengths. \(a=1, \quad b=2, \quad c=2\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 0.968 square units.

Step 1: Check if a triangle can be formed

To determine if a triangle is possible with sides of length 1, 2, and 2, we use the triangle inequality theorem. The sum of the lengths of any two sides must be greater than the length of the third side. We check the following inequalities: \(1 + 2 > 2\), \(1 + 2 > 2\), and \(2 + 2 > 1\). All are true, so a triangle can be formed.

Step 2: Identify the type of triangle

Given the sides are \(1, 2,\) and \(2\), this triangle is isosceles because it has two equal sides.

Step 3: Use Heron's formula to find the area

First, calculate the semi-perimeter \(s\) of the triangle using the formula \[s = \frac{a + b + c}{2}\] Substituting the side lengths gives \[s = \frac{1 + 2 + 2}{2} = \frac{5}{2} = 2.5\]Next, use Heron's formula for the area \(A\):\[A = \sqrt{s(s-a)(s-b)(s-c)}\]Substitute in the values:\[A = \sqrt{2.5(2.5-1)(2.5-2)(2.5-2)}\] Calculate:\[A = \sqrt{2.5 \times 1.5 \times 0.5 \times 0.5} = \sqrt{2.5 \times 0.375} = \sqrt{0.9375} \approx 0.968\]

Step 4: Conclude with the solution

The area of the triangle with sides of lengths 1, 2, and 2 is approximately 0.968 square units.

Key Concepts

Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental rule in geometry. It determines whether three lengths can form a triangle. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If any of these conditions aren't met, the lines won't connect to form a closed shape.
For example, let's consider three sides with lengths 1, 2, and 2. You apply the triangle inequality theorem by checking the following:

• The sum of the first and second sides (1 + 2) must be greater than the third side (2).
• The sum of the first and third sides (1 + 2) must also be greater than the second side (2).
• Finally, the sum of the second and third sides (2 + 2) must be greater than the first side (1).

All these conditions are satisfied, so these lengths can indeed form a triangle. Understanding this theorem is crucial. It helps determine whether a set of side lengths can create a valid triangle. If any one condition fails, you simply cannot form a triangle with those lengths.

Heron's Formula

Heron's Formula is an elegant way to calculate the area of a triangle when you know the lengths of all three sides. This makes it especially useful for triangles that aren't right-angled. To use Heron's Formula, you start by finding the semi-perimeter of the triangle, denoted as \( s \). The semi-perimeter is half of the sum of the three sides:

• Calculate the semi-perimeter: \( s = \frac{a+b+c}{2} \)
• For sides 1, 2, and 2, the semi-perimeter is \( s = \frac{1+2+2}{2} = 2.5 \)

Once you have \( s \), plug it into the formula for the area \( A \):\[ A = \sqrt{s(s-a)(s-b)(s-c)} \] In this case, substituting gives \( A = \sqrt{2.5 \times (2.5-1) \times (2.5-2) \times (2.5-2)} \).
Carefully carry out the multiplications and square root to find the area, which will be approximately 0.968 square units. Heron's Formula is particularly informative because it provides an exact area for any triangle, not just special types like right triangles.

Isosceles Triangle

An Isosceles Triangle is a type of triangle that has at least two sides of equal length. This makes it special because it has a unique symmetry and certain predictable properties. For the sides given as 1, 2, and 2, this triangle qualifies as isosceles because two of its sides are equal (2 and 2).
Recognizing an isosceles triangle is helpful because:

• It has two equal sides, which often means two equal angles opposite those sides.
• This characteristic can be used to deduce properties about the triangle, such as angles and area.
• Understanding its symmetry can make solving problems involving this type of triangle easier.

Isosceles triangles are frequently encountered and understanding their properties can simplify many geometry problems. By knowing two sides are equal, it can be easier to apply formulas and make calculations.