Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Heron Triangles A Heron triangle is a triangle having integer sides and integer area. Show that each of the following is a Heron triangle. A. \(a=11, b=13, c=20\) B. \(a=13, b=14, c=15\) C. \(a=7, b=15, c=20\)

Step-by-Step Solution

Answer

Triangles A, B, and C are Heron triangles as they have integer side lengths and integer areas.

Step 1: Calculate semi-perimeter of triangle A

For triangle A with sides \(a=11\), \(b=13\), and \(c=20\), first calculate the semi-perimeter \(s\) using \(s = \frac{a+b+c}{2}\). Here, \(s = \frac{11+13+20}{2} = 22\).

Step 2: Use Heron's formula for triangle A

Using Heron's formula, the area \(A\) is given by \( A = \sqrt{s(s-a)(s-b)(s-c)} \). Substitute the values, \( A = \sqrt{22(22-11)(22-13)(22-20)} = \sqrt{22 \times 11 \times 9 \times 2} = \sqrt{4356} = 66 \). The area is an integer.

Step 3: Calculate semi-perimeter of triangle B

For triangle B with sides \(a=13\), \(b=14\), and \(c=15\), calculate the semi-perimeter \(s = \frac{13+14+15}{2} = 21 \).

Step 4: Use Heron's formula for triangle B

For triangle B, the area \(A\) is \( A = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84 \). The area is an integer.

Step 5: Calculate semi-perimeter of triangle C

For triangle C with sides \(a=7\), \(b=15\), and \(c=20\), calculate the semi-perimeter \(s = \frac{7+15+20}{2} = 21 \).

Step 6: Use Heron's formula for triangle C

For triangle C, the area \(A\) is \( A = \sqrt{21(21-7)(21-15)(21-20)} = \sqrt{21 \times 14 \times 6 \times 1} = \sqrt{1764} = 42 \). The area is an integer.

Key Concepts

Semi-perimeter

The semi-perimeter of a triangle is a crucial step in applying Heron's formula. It is half the perimeter of the triangle, and is represented by the symbol \( s \). To find the semi-perimeter, you add the lengths of all three sides of the triangle and then divide by two. This formula can be expressed as:
\[ s = \frac{a+b+c}{2} \]
where \( a \), \( b \), and \( c \) are the side lengths of the triangle.

• For Triangle A with sides \( 11, 13, \) and \( 20 \), the semi-perimeter \( s = \frac{11+13+20}{2} = 22 \).
• For Triangle B with sides \( 13, 14, \) and \( 15 \), \( s = \frac{13+14+15}{2} = 21 \).
• For Triangle C with sides \( 7, 15, \) and \( 20 \), \( s = \frac{7+15+20}{2} = 21 \).

Calculating the semi-perimeter is the first step in determining the area of a triangle using Heron's formula. It simplifies the next steps when calculating each segment's contribution towards the area.

Heron's formula

Heron's formula is a method to find the area of a triangle when you know the length of all three sides. It's particularly useful because it allows you to calculate the area without needing the triangle's height. The formula is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( s \) is the semi-perimeter, and \( a \), \( b \), and \( c \) are the triangle's sides.

• For Triangle A, with a semi-perimeter of \( 22 \), the area is calculated as:
  \( A = \sqrt{22(22-11)(22-13)(22-20)} = \sqrt{22 \times 11 \times 9 \times 2} = 66 \).
  The area is an integer.
• For Triangle B, using the semi-perimeter \( 21 \), the area is:
  \( A = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = 84 \).
  The area, again, is an integer.
• For Triangle C, also using \( 21 \) as the semi-perimeter, the area is:
  \( A = \sqrt{21(21-7)(21-15)(21-20)} = \sqrt{21 \times 14 \times 6 \times 1} = 42 \).
  As expected, the area is an integer.

By applying the semi-perimeter and known side measures, Heron's formula quickly confirms the integer areas of Heron triangles.

Integer Area

When a triangle has an integer area, it implies that both the configuration of its sides and their arrangement allow for a perfect square calculation under Heron's formula. Often referred to as a Heron triangle when all sides and the area are integers, these triangles are relatively unique and significant in geometry.

• A triangle is considered a Heron triangle if:
  - All side lengths \( a, b, c \) are integers. - The area calculated using Heron’s formula is also an integer.
• For example: - Triangle A with sides 11, 13, and 20 results in area \( 66 \). - Triangle B, with sides 13, 14, and 15, results in area \( 84 \). - Triangle C, with sides 7, 15, and 20, results in area \( 42 \).

Finding such triangles involves working through Heron’s formula to see if the end result is an integer. These provide excellent practical exercises in geometry, emphasizing not just calculation, but understanding of how side lengths correlate with area.