Problem 85 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Perfect Triangles A perfect triangle is a triangle whose sides have whole- number lengths and whose area is numerically equal to its perimeter. Show that the triangle with sides of lengths \(9,10,\) and 17 is perfect.

Step-by-Step Solution

Answer

The triangle with sides 9, 10, and 17 is perfect because its area (36) equals its perimeter (36).

Step 1: Understanding the Perimeter

First, find the perimeter of the triangle. The perimeter is the sum of the lengths of the sides. For a triangle with sides 9, 10, and 17, the perimeter is:\[ P = 9 + 10 + 17 = 36 \]

Step 2: Calculating the Area Using Heron's Formula

Next, use Heron's Formula to find the area of the triangle. First, calculate the semi-perimeter \(s\):\[ s = \frac{P}{2} = \frac{36}{2} = 18 \]Now, apply Heron’s Formula:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute the side lengths and semi-perimeter:\[ A = \sqrt{18(18-9)(18-10)(18-17)} = \sqrt{18 \cdot 9 \cdot 8 \cdot 1} \]Calculate:\[ A = \sqrt{1296} = 36 \]

Step 3: Comparison of Area and Perimeter

To verify if the triangle is a perfect triangle, compare the area to the perimeter. In this case, the area calculated is 36, and the perimeter is also 36. Since both are numerically equal, this confirms the triangle is a perfect triangle.

Key Concepts

Heron's Formula for Area Calculation

Heron's Formula is a valuable tool used to find the area of a triangle when you know the lengths of all its sides. Unlike other methods, you don't need to know any angles or other dimensions. Here's a simple way to use it:

• First, calculate the semi-perimeter of the triangle, which is half of the perimeter.
• Next, apply the formula: \[A = \sqrt{s(s-a)(s-b)(s-c)}\]where \(A\) is the area, \(s\) is the semi-perimeter, and \(a, b, c\) are the side lengths of the triangle.
• Plug in your measurements and solve for the area.

For the triangle with sides of 9, 10, and 17, the semi-perimeter \(s = 18\). Plugging these into Heron's Formula provides us with \(A = 36\), demonstrating a perfect balance where area equals the perimeter, confirming the triangle is perfect.

Understanding Triangle Perimeter

The perimeter of a triangle is the total length around it, simply the sum of its three sides. Calculating the perimeter is straightforward:

• Add together the lengths of each side.
• For example, with sides of 9, 10, and 17, the perimeter \(P = 9 + 10 + 17 = 36\).
• This calculated perimeter is particularly important for identifying perfect triangles, as it must equal the numerical value of the area.

Knowing the perimeter also helps when using Heron's Formula, as it leads to the next step: finding the semi-perimeter.

Basic Area Calculation Concepts

Calculating the area of a triangle depends on having the right measurements. However, with Heron's Formula, the calculation is simplified and can be performed without additional angle measurements.

• Use the semi-perimeter obtained from the perimeter.
• Substitute into the Heron’s Formula to determine the area.
• Even without side angles or heights, this formula efficiently gives you the area.

Perfect triangles offer a unique case where the area and perimeter are numerically matched. This gives you a perfect numerical harmony, as seen in the area of 36 for sides 9, 10, and 17, matching the perimeter.

Right Triangle Properties

Right triangles have their own set of specific properties, useful in various calculations, but it's important to note that having whole-number side lengths meeting both area and perimeter criteria doesn't always mean a triangle is a right triangle.

• A right triangle features one angle of 90 degrees plus a hypotenuse, the longest side.
• Calculations often incorporate the Pythagorean Theorem: \[c^2 = a^2 + b^2\]where \(c\) is the hypotenuse.
• Not all perfect triangles are right triangles, as demonstrated in this case, where a non-right perfect triangle still shares numerical perimeter and area values.

Understanding properties such as the hypotenuse or angle measures can enhance comprehension, though Heron's simplicity with area doesn't require them in such non-right triangle situations.