SOLUTION: One side of a rectangle is 7 cm longer than the other side. The diagonal is 8 cm longer than the shortest side of the rectangle. Find the dimensions of the rectangle. I am stuck an

Algebra ->  Rectangles -> SOLUTION: One side of a rectangle is 7 cm longer than the other side. The diagonal is 8 cm longer than the shortest side of the rectangle. Find the dimensions of the rectangle. I am stuck an      Log On


   

Question 4129: One side of a rectangle is 7 cm longer than the other side. The diagonal is 8 cm longer than the shortest side of the rectangle. Find the dimensions of the rectangle. I am stuck and do not know how to solve this problem
Found 2 solutions by ivankst, rapaljer:
Answer by ivankst(3) About Me  (Show Source):
You can put this solution on YOUR website!
Let a and b be sides of a rectangle, and d a diagonal. Let b%3Ca.
We have following equations:
a=b%2B7
d=b%2B8
By using Pythagorean theorem you can calculate that d=sqrt%28a%5E2%2Bb%5E2%29
So our system of equations becomes:
a=b%2B7
sqrt%28a%5E2%2Bb%5E2%29=b%2B8
If we put a=b%2B7 in second equation, we get:
sqrt%28%28b%2B7%29%5E2%2Bb%5E2%29=b%2B8
%28b%2B7%29%5E2%2Bb%5E2=%28b%2B8%29%5E2
b%5E2%2B14b%2B49%2Bb%5E2=64%2B16b%2Bb%5E2
b%5E2-2b-15=0
b=%282%2B-sqrt%284-4%2A1%2A%28-15%29%29%29%2F2
b=%282%2B-sqrt%2864%29%29%2F2
b=%282%2B-8%29%2F2
b=5 or b=-3
b cannot be a negative number, so b=5.
First eqaution gives us a:
a=b%2B7
a=5%2B7
a=12
So, sides of a rectangle are: (5,12).

Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = first side of the rectangle (width)
x+7 = second side of the rectangle (length)
x+8 = diagonal of the rectangle (hypotenuse of the right triangle)
x%5E2+%2B+%28x%2B7%29%5E2+=+%28x%2B8%29%5E2+
+x%5E2+%2B+x%5E2+%2B+14x+%2B+49+=+x%5E2+%2B+16x+%2B+64+
This is a quadratic equation, so set it equal to zero!
+x%5E2+%2B+x%5E2+%2B+14x+%2B+49+-+x%5E2+-+16x+-+64+=+0
x%5E2+-+2x+-+15+=+0
Factor the left side:
+%28x+-+5%29%28x+%2B+3%29+=+0+
x = 5 or x = -3
Since x is a side of a triangle, it cannot be negative. Therefore reject the x=-3.
If x = 5, then x + 7 = 12, and x + 8 = 13.
The rectangle is 5 by 12 and the diagonal is 13.