SOLUTION: The diagonal of a rectangle is 2 ft longer than its length and 5 feet longer than its width. What are the dimensions of the rectangle?

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Question 107935: The diagonal of a rectangle is 2 ft longer than its length and 5 feet longer than its width. What are the dimensions of the rectangle?
Found 2 solutions by MathLover1, edjones:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
The diagonal of a rectangle is 2+ft longer than its length and 5+ft longer than its width.
What are the dimensions of the rectangle?
Since the diagonal divides rectangle into two righttrianles, we will use Pythagoras%27+Theorem+ in order to find the dimensions of the rectangle.
In algebraic terms, a2+%2B+b2+=+c2where c is the hypotenuse ( in your case the diagonal) while a and b+are the legs of the triangle (in your case the lenght and width).
given:
c+=+a+%2B+2ft => a+=+c+%96+2ft
c+=+b+%2B+5ft => b+=+c+%96+5ft

we will substitute it in:
c%5E2+=+a%5E2+%2B+b%5E2
c%5E2+=+%28c-2ft%29%5E2+%2B+%28c-5ft%29%5E2
c%5E2+=+c%5E2+-4cft+%2B+4ft%5E2+%2B+c%5E2+%96+10cft+%2B+25ft%5E2
c%5E2+=+2c%5E2+%96+14cft+%2B29ft%5E2
c%5E2+-14cft+%2B+29ft%5E2+=+0
c%5B1%2C2%5D=%2814+%2B-+sqrt+%2814%5E2+-4%2A1%2A29+%29%29+%2F+%282%2A1%29
We need only positive root
c%5B1%5D=%2814+%2B+sqrt+%2814%5E2+-4%2A1%2A29+%29%29+%2F+%282%2A1%29

c%5B1%5D=%2814+%2B+sqrt+%28196-116++%29%29+%2F+2
c%5B1%5D=%2814+%2B+sqrt+%2880%29%29+%2F+2

c%5B1%5D=%2814+%2B+8.9%29+%2F+2
c%5B1%5D=%2822.9%29+%2F+2
c%5B1%5D=%2811.45ft%29+


a+=+c+%96+2ft
=> a+=+11.45ft+%96+2ft+=+9.45ft
b+=+c+%96+5ft
b+=+11.45ft+%96+5ft
=> b+=+6.45ft
Check:
%2811.45%29%5E2+=+%289.45%29%5E2+%2B+%286.45%29%5E2
131.1025+=+89.3025+%2B+41.6025
131+=+131

Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
The diagonal divides the rectangle into 2 equal right triangles.
Let the diagonal=c and the 2 sides be a and b.
a=c-2, b=c-5
a^2+b^2=c^2
(c-2)^2+(c-5)^2=c^2
c^2-4c+4+c^2-10c+25=c^2
c^2-14c+29=0
c^2-14c =-29
c^2-14c+49=49-29 complete the square
(c-7)^2=20
c-7=+-sqrt(20)
c=7+-sqrt(5*4)
=7+-2sqrt(5)
Only 7+2sqrt(5) is the only correct answer for the diagonal. the other would lead to negative numbers for the sides of the rectangle.
The sides of the rectangle are:
5+2sqrt(5) and 2+2sqrt(5)
Ed