find the length of each boundary
Vista county is setting aside a large parcel of land to preserve
it as open space. the county has hired Jane's surveying firm to
survey the parcel, which is in the shape of a right triangle. the
longer leg of the triangle measures 5 miles less than the square of
the shorter leg, and the hypotenuse of the triangle measures 13 miles
less than twice the square of the shorter leg. the length of each
boundary is a whole number. find the length of each boundary.
Make a the long side, b the short side and c the hypotenuse.
a = b^2 - 5
b = b, obviously
c = 2b^2 - 13
From the Pythagorean theorem, we have a^2 + b^2 = c^2
(b^2 - 5)^2 + b^2 = (2b^2 - 13)^2
(b^2 - 5) * (b^2 - 5) + b^2 = (2b^2 - 13) * (2b^2 - 13)
b^4 - 5b^2 - 5b^2 + 25 + b^2 = 4b^4 - 26b^2 - 26b^2 + 169
b^4 - 9b^2 + 25 = 4b^4 - 52b^2 + 169
Multiply by -1. That way, when the terms on the right side are
moved to the left side, the b^4 coefficient will be positive.
-b^4 + 9b^2 - 25 = -4b^4 + 52b^2 - 169
-b^4 + 4b^4 + 9b^2 - 52b^2 - 25 + 169 = 0
3b^4 - 43b^2 + 144 = 0
We need to factor the left side. If we use 12 and 12 as the factors
for 144, we don't get the required coefficient for the b^2 term.
The factors 16 and 9 will work because we multiply the 9 by 3.
(3b^2 - 16)(b^2 - 9) = 0
Using those factors we have
(3b^2 -16) = 0
or
(b^2 - 9) = 0
Solving for b in the first factor:
(3b^2 -16) = 0
3b^2 = 16
b^2 = 16/3 = 5.33333
b = 2.3094
This doesn't satisfy the requirement that all lengths are whole numbers.
Solving for b in the second factor:
(b^2 - 9) = 0
b^2 = 9
b = 3
Actually, b = -3 is also valid, but the length cannot be negative.
We have side b = 3 miles,
a = b^2 - 5 = 9 - 5 = 4 miles,
c = 2b^2 - 13 = 2(9) - 13 = 18 - 13 = 5 miles
So, we have a simple 3-4-5 right triangle.