Answer:
A.) The greatest possible distance between any of the stakes is 53 feet.
B.) A triangle with sides 15, 36, and 39 is a right triangle and has perimeter 90 feet.
Step-by-step explanation:
Given - Fiona wants to rope off a triangular region in her yard for a vegetable garden. She has 90 feet of rope to use, she wants the distance between each stake to be an integer, and the distance between two of the stakes must be 36 feet.
To find - A. What is the greatest possible distance between any of the stakes?
B. If she wants to create a right triangular region and use all of the 90 feet of rope, what would be the lengths of the sides of the region?
Solution -
A.)
Let us assume that,
Length of side 1 = x
Length of side 2 = y
Length of side 3 = z
Now,
Given that,
She has 90 feet of rope to use
⇒x + y + z = 90
And
The distance between two of the stakes must be 36 feet.
So, Let z = 36.
∴ we get
x + y + 36 = 90
⇒x + y = 90 - 36
⇒x + y = 54
Now,
Let x be as large as possible.
i.e. To make x to be large, y has to be as small as possible.
It happens when y = 1.
( We can not take y = 0, or else a triangle won't form)
∴ we get
y = 1
⇒x + 1 = 54
⇒x = 54 - 1
⇒x = 53
∴ we get
The greatest possible distance between any of the stakes is 53 feet.
B.)
We know that,
(5,12,13) is a Pythagorean triple.
Because 5² + 12² = 13²
So,
As, we have a Pythagorean triple,
This means a triangle with sides 5, 12, 13 is a right triangle.
Now,
If a triangle has sides 5, 12, 13,
⇒ The perimeter becomes 5 + 12 + 13 = 30.
By Triple each side we get,
Length of sides 15, 36, 39.
The perimeter i= 15 + 36 + 39 = 90
Also,
We can see that,
15² + 36² = 39².
This means we have a right triangle and (15, 36, 39) is another Pythagorean triple.
Because any scaled version of a Pythagorean triple, is also a Pythagorean triple.
∴ we get
A triangle with sides 15, 36, and 39 is a right triangle and has perimeter 90 feet.
