One of two numbers that can be both the area and perimeter of a triangle whose side lengths are a Pythagorean triple?
It is here...
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html#patterns
One of two numbers that can be both the area and perimeter of a triangle whose side lengths are a Pythagorean triple.
The 5-12-13 Pythagorean Triple triangle has an area and perimeter of 30.
To find the number that can be both the area and perimeter of a triangle with side lengths as Pythagorean triples, let's first understand what a Pythagorean triple is.
A Pythagorean triple consists of three positive integers (a, b, c) that satisfy the Pythagorean theorem: a^2 + b^2 = c^2. In the context of a triangle, 'a' and 'b' represent the lengths of the two shorter sides (also known as the legs), and 'c' represents the length of the longest side (also known as the hypotenuse).
The perimeter of the triangle is the sum of the lengths of all three sides, which can be written as a + b + c.
The area of a triangle can be calculated using the formula: Area = (1/2) * base * height. In the case of a right-angled triangle, the legs 'a' and 'b' are considered the base and height, respectively.
Now, let's find the number that satisfies both the area and perimeter conditions of the triangle.
Since the perimeter of the triangle is a + b + c, and the area is (1/2) * a * b, we can set up the following equation:
a + b + c = (1/2) * a * b
To simplify the equation, we can multiply every term by 2 to eliminate the fraction:
2a + 2b + 2c = a * b
Next, let's explore various Pythagorean triple combinations to find a solution:
1. (3, 4, 5) triple:
Using these values, we get: 2(3) + 2(4) + 2(5) = (3)(4)
Simplifying further: 6 + 8 + 10 = 12
This equation does not satisfy our conditions.
2. (5, 12, 13) triple:
Using these values, we get: 2(5) + 2(12) + 2(13) = (5)(12)
Simplifying further: 10 + 24 + 26 = 60
This equation does not satisfy our conditions either.
3. (6, 8, 10) triple:
Using these values, we get: 2(6) + 2(8) + 2(10) = (6)(8)
Simplifying further: 12 + 16 + 20 = 48
This equation satisfies both the perimeter and area conditions.
Therefore, in the case of a Pythagorean triple, the number that can be both the area and perimeter of the triangle is 48.