Problem 50 - Precalculus

Question

Find the area of each triangle with measures given. $$a=19, b=23, c=3$$

Step-by-Step Solution

Answer

The lengths do not form a triangle; no area exists.

Step 1: Calculate the Semiperimeter

First, compute the semiperimeter (s) of the triangle. The semiperimeter is half the perimeter of the triangle. The formula for the semiperimeter is:\[ s = \frac{a + b + c}{2} \]Substitute the given values:\[ s = \frac{19 + 23 + 3}{2} = \frac{45}{2} = 22.5 \]

Step 2: Apply Heron's Formula

Now that we have the semiperimeter, use Heron's formula to find the area of the triangle. Heron's formula is given by:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute the values calculated:\[A = \sqrt{22.5(22.5 - 19)(22.5 - 23)(22.5 - 3)}\]This calculation will result in the following terms:\[ A = \sqrt{22.5 \times 3.5 \times (-0.5) \times 19.5}\]

Step 3: Recognize the Impossibility of the Triangle

The calculation reveals a multiplication involving a negative number (due to \((s-b)\) being negative), indicating that a triangle cannot have these dimensions. In any valid triangle, each side must be shorter than the sum of the other two sides. Check if this condition is violated by one of the sides:\[ a + b = 19 + 23 = 42 > 3 \]\[ a + c = 19 + 3 = 22 < 23 \] \[ b + c = 23 + 3 = 26 < 19 \]Both \(a + c\) and \(b + c\) are less than the third side, indicating the given lengths do not form a triangle.

Key Concepts

Understanding the Semiperimeter

The semiperimeter is an essential concept in geometry, especially when working with Heron's formula to determine the area of a triangle. It represents half the perimeter of the triangle. When calculating the semiperimeter, you add up all the sides of the triangle and then divide that sum by two. The formula for the semiperimeter is simple:

• \( s = \frac{a+b+c}{2} \)

In the given problem, substituting the measures \( a = 19 \), \( b = 23 \), and \( c = 3 \) into the formula, we calculated

• \( s = \frac{19 + 23 + 3}{2} = 22.5 \)

This semiperimeter is vital for applying Heron's formula. However, it's also key for understanding if a set of sides can form a triangle, as seen here.

Exploring the Triangle Inequality

The triangle inequality is a crucial principle that determines if three given lengths can actually form a triangle. According to this rule, for any three sides \( a \), \( b \), and \( c \), the sum of the lengths of any two sides must be greater than the length of the remaining side. This can be expressed in three inequalities:

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

In the exercise, the side lengths were given as \( a = 19 \), \( b = 23 \), \( c = 3 \). When checking these inequalities:

• \( a + b = 42 > 3 \)
• \( a + c = 22 < 23 \)
• \( b + c = 26 < 19 \)

The inequalities \( a + c > b \) and \( b + c > a \) are not satisfied, indicating these sides cannot form a valid triangle. This step is often overlooked but is fundamental in confirming the validity of using geometric formulas.

Triangle Area Calculation Using Heron's Formula

Heron's formula is a handy tool for calculating the area of a triangle when you know the lengths of all three sides. The formula is:

• \( A = \sqrt{s(s-a)(s-b)(s-c)} \)

Once the semiperimeter \( s \) is determined, you substitute the values into the formula. In this exercise, substituting reveals an issue because one of the

• terms \((s-b)\) becomes negative \((-0.5)\)

This signals that a valid triangle doesn't exist with these side lengths. Attempting to calculate further will result in an imaginary number, underscoring the necessity of first checking the triangle inequality before applying Heron's formula. This demonstrates how the formula and triangle inequality work together to validate problem-solving in geometry.