Problem 55 - Algebra and Trigonometry

Question

A perfect triangle is one having integers for sides for which the area is numerically equal to the perimeter. Show that the triangles with the given side lengths are perfect. (a) 9,10,17 (b) 6,25,29

Step-by-Step Solution

Answer

Both triangles (9,10,17) and (6,25,29) are perfect triangles as their area is numerically equal to their perimeter.

Step 1 - Verify the sides form a triangle

To confirm that the given sides form a valid triangle, use the triangle inequality theorem. For any three sides of a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Check this for both sets of side lengths.

Step 2 - Calculate the Perimeter

The perimeter of a triangle is the sum of its side lengths. For triangle (a), the perimeter is calculated as follows: \[ P = 9 + 10 + 17 = 36 \] For triangle (b), the perimeter calculation is: \[ P = 6 + 25 + 29 = 60 \]

Step 3 - Calculate the Area using Heron's Formula

Use Heron's formula to find the area of the triangles. Heron's formula states that the area of a triangle whose sides are of length \(a\), \(b\), and \(c\) is given by \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \(s\) is the semiperimeter of the triangle, \(s = \frac{P}{2}\).For triangle (a) with sides 9, 10, and 17: \[ s = \frac{36}{2} = 18 \] \[ A = \sqrt{18(18-9)(18-10)(18-17)} = \sqrt{18 \times 9 \times 8 \times 1} = \sqrt{1296} = 36 \]For triangle (b) with sides 6, 25, and 29: \[ s = \frac{60}{2} = 30 \] \[ A = \sqrt{30(30-6)(30-25)(30-29)} = \sqrt{30 \times 24 \times 5 \times 1} = \sqrt{3600} = 60 \]

Step 4 - Compare Area and Perimeter

Finally, compare the area and perimeter for both triangles. If the area is numerically equal to the perimeter, then the triangle is a perfect triangle.For triangle (a): Perimeter = 36 Area = 36 Since Area = Perimeter, triangle (a) is a perfect triangle.For triangle (b): Perimeter = 60 Area = 60 Since Area = Perimeter, triangle (b) is a perfect triangle.

Key Concepts

triangle inequality theorem

The triangle inequality theorem is fundamental in determining whether three given sides can form a triangle. According to this theorem, the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. For example, consider sides 9, 10, and 17.
We need to check the following conditions:

• 9 + 10 > 17
• 9 + 17 > 10
• 10 + 17 > 9

All conditions are satisfied. Thus, these sides can form a triangle. Repeat this check for sides 6, 25, and 29:

• 6 + 25 > 29
• 6 + 29 > 25
• 25 + 29 > 6

Again, all conditions are met, so the sides 6, 25, and 29 can form a triangle.

perimeter

The perimeter of a triangle is the total distance around the triangle, which is simply the sum of its three sides. For instance, for a triangle with sides 9, 10, and 17, the perimeter (P) is:

• P = 9 + 10 + 17 = 36

Similarly, for a triangle with sides 6, 25, and 29, the perimeter (P) is:

• P = 6 + 25 + 29 = 60

This step is essential for comparing the perimeter with the area of the triangle later.

Heron's formula

Heron's formula is used to find the area of a triangle when all three sides are known. According to Heron's formula, the area (A) of a triangle with sides of lengths a, b, and c is:
{ $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ } where 's' is the semi-perimeter of the triangle and is calculated as:
{$$s = \frac{P}{2}$$}
For instance, for a triangle with sides 9, 10, and 17, we first calculate the semi-perimeter (s): {$$s = \frac{36}{2} = 18$$} Then, we apply Heron's formula: {$$A = \sqrt{18(18-9)(18-10)(18-17)} = \sqrt{18 \times 9 \times 8 \times 1} = \sqrt{1296} = 36$$}
Similarly, for a triangle with sides 6, 25, and 29, the semi-perimeter is calculated as follows: $$s = \frac{60}{2} = 30$$
Applying Heron's formula to find the area: $$A = \sqrt{30(30-6)(30-25)(30-29)} = \sqrt{30 \times 24 \times 5 \times 1} = \sqrt{3600} = 60$$

semiperimeter

The semiperimeter of a triangle is half of its perimeter and is denoted by 's'. It is used in Heron's formula to help calculate the area of the triangle more easily.
To find the semiperimeter, use the formula: {$$s = \frac{P}{2}$$}
For a triangle with a perimeter of 36, the semiperimeter is calculated as: $$s = \frac{36}{2} = 18$$ For another triangle with a perimeter of 60, the semiperimeter is: $$s = \frac{60}{2} = 30$$
This semiperimeter value is crucial in determining the area using Heron's formula.

area of a triangle

The area of a triangle is a measure of the space enclosed by the triangle. For triangles with sides of known lengths, we typically use Heron's formula to find the area. As demonstrated earlier: For a triangle with sides 9, 10, and 17, the area (A) is calculated as: {$$A = \sqrt{18(18-9)(18-10)(18-17)} = \sqrt{1296} = 36$$}
Similarly, for a triangle with sides 6, 25, and 29, the area is: {$$A = \sqrt{30(30-6)(30-25)(30-29)} = \sqrt{3600} = 60$$}
In both examples above, the area matches the perimeter, making these triangles perfect triangles according to the problem's definition.