SOLUTION: Section 9.6 # 45 Geometry. The length of one leg of a right triangle is 3 in. more than the other.If the length of the hypotenuse is 15 in., what are the lengths of the two le

Algebra ->  Algebra  -> Triangles -> SOLUTION: Section 9.6 # 45 Geometry. The length of one leg of a right triangle is 3 in. more than the other.If the length of the hypotenuse is 15 in., what are the lengths of the two le      Log On

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Question 73992: Section 9.6 # 45

Geometry. The length of one leg of a right triangle is 3 in. more than the other.If the length of the hypotenuse is 15 in., what are the lengths of the two legs?

Found 2 solutions by jim_thompson5910, checkley75:
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
The word problem translates to
a=3%2Bb"...one leg (a) is (=) 3 more (+) than another leg (b)"
So plug this into Pythagoreans theorem
a%5E2%2Bb%5E2=c%5E2where a and b are the legs and c is the hypotenuse.
%283%2Bb%29%5E2%2Bb%5E2=15%5E2Plug in 3+b into a, this eliminates a
9%2B6b%2B2b%5E2=225Add like terms and simplify.
2b%5E2%2B6b%2B9-225=0Subtract 225 from both sides
2b%5E2%2B6b-216=0
Now use the quadratic formula to solve for b
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ab%5E2%2Bbb%2Bc=0 (in our case 2b%5E2%2B6b%2B-216+=+0) has the following solutons:

b%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%286%29%5E2-4%2A2%2A-216=1764.

Discriminant d=1764 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-6%2B-sqrt%28+1764+%29%29%2F2%5Ca.

b%5B1%5D+=+%28-%286%29%2Bsqrt%28+1764+%29%29%2F2%5C2+=+9
b%5B2%5D+=+%28-%286%29-sqrt%28+1764+%29%29%2F2%5C2+=+-12

Quadratic expression 2b%5E2%2B6b%2B-216 can be factored:
2b%5E2%2B6b%2B-216+=+2%28b-9%29%2A%28b--12%29
Again, the answer is: 9, -12. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B6%2Ax%2B-216+%29

Disregard the negative answer (a negative length doesn't make any sense). So the length of one leg is 9 in. Use this to find the other leg.
a=3%2Bb
a=3%2B9
a=12
So the other leg is 12 in.


Check:
a%5E2%2Bb%5E2=c%5E2
12%5E2%2B9%5E2=15%5E2
144%2B81=225
225=225Works


Answer by checkley75(3666) About Me  (Show Source):
You can put this solution on YOUR website!
x^2+(x+3)^2=15^2
x^2+x^2+6x+9=225
2x^2+6x+9-225=0
2x^2+6x-216=0
x^2+3x-108=0
(x-9)(x+12)=0
x-9=0
x=9 answer for side 1.
x+12=0
x=-12 not an answer.
9+3=12 is the answer for side 2.
proof
9^2+12^2=225
81+144=225
225=225