SOLUTION: A garden lot is in the shape of a right triangle. The hypothenuse is 30 yds. longer than twice the length of the shorter leg, and the longer leg is 20 yds. longer than twice the le

Algebra ->  Pythagorean-theorem -> SOLUTION: A garden lot is in the shape of a right triangle. The hypothenuse is 30 yds. longer than twice the length of the shorter leg, and the longer leg is 20 yds. longer than twice the le      Log On


   

Question 173498: A garden lot is in the shape of a right triangle. The hypothenuse is 30 yds. longer than twice the length of the shorter leg, and the longer leg is 20 yds. longer than twice the length of the shorter leg. What are the lengths of the three sides?
Found 2 solutions by solver91311, nerdybill:
Answer by solver91311(24713) About Me  (Show Source):
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Short leg: x
Long leg: 2x%2B20
Hypotenuse: 2x%2B30

Pythagoras says:

x%5E2%2B%282x%2B20%29%5E2=%282x%2B30%29%5E2

Expand the binomials:

x%5E2%2B4x%5E2%2B80x%2B400=4x%5E2%2B120x%2B900

Add -4x%5E2-120x-900 to both sides:

x%5E2%2B4x%5E2-4x%5E2%2B80x-120x%2B400-900=0

Collect terms:

x%5E2-40x-500=0

Solve the quadratic. Hint: %2810%29%28-50%29=-500 and 10%2B%28-50%29=-40

One of your roots will be negative. This is an extraneous root created when we squared the variable in the process of solving the problem, so discard it. The positive root will be the measure of the shorter leg. The other two length measures should be simple to calculate from the relationships in the problem givens.





Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
Apply the Pythagorean theorem:
Let x = shorter leg
then
30 + 2x = hypotenuse
20 + 2x = longer leg
.
From Pythagorean we have:
(30 + 2x)^2 = x^2 + (20 + 2x)^2
900 + 120x + 4x^2 = x^2 + 400 + 80x + 4x^2
900 + 120x = x^2 + 400 + 80x
500 + 120x = x^2 + 80x
500 + 40x = x^2
0 = x^2 -40x + 500
0 = (x-50)(x+10)
.
x = {50, -10}
.
Throw out the negative solution leaving us with:
x = 50 yards (shorter leg)
.
hypotenuse:
30 + 2x = 30 + 2(50) = 130 yards
.
Longer leg:
20 + 2x = 20 + 2(50) = 120 yards