Problem 1
- a = length of side 1
- b = length of side 2
- c = length of side 3
Let's say that c = 36.
Since she has 90 feet of rope total, that leaves 90-36 = 54 feet of rope for sides 'a' and b to divide up somehow. We can say a+b = 54
Now let's say we want 'a' to be as large as possible. To do this, we need to make b as small as possible. That occurs when b = 1. We can't have b = 0, or else a triangle won't form. So the next value up is b = 1.
If b = 1, then,
a+b = 54
a = 54-b
a = 54-1
a = 53
The longest side possible is 53 feet
<h3>Answer: 53 feet</h3>
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Problem 2
Recall that (5,12,13) is a pythagorean triple. This is because 5^2+12^2 = 13^2. Both sides lead to 169.
Since we have a pythagorean triple, this indicates a triangle with sides 5,12,13 is a right triangle.
If a triangle has sides 5,12, and 13, then the perimeter is 5+12+13 = 30.
Triple each side to get a larger triangle of 15, 36, and 39. The larger perimeter is 90 after adding those larger sides. Note the jump from 30 to 90 is "times 3". It's not a coincidence that the perimeter has multiplied by the same scale factor as the side lengths.
Also note that 15^2+36^2 = 39^2. Both sides lead to 1521. This indicates we have a right triangle and (15,36,39) is another pythagorean triple. Any scaled version of a pythagorean triple, is also a pythagorean triple.
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In short, we've found that a triangle with sides 15, 36, and 39 is a right triangle and has perimeter 90 feet.
<h3>Answer: 15 ft, 36 ft, 39 ft</h3>
