Pythagorean theorem?

Question

I have a question that asks to find the length and width of an object with an area of 108 sq feet and a diagonal mesurement of 15 feet. I know it needs to be solved as a system. This is what I have
xy=108
x^2+y^2=15 (should it be ^2 also?? or is it already?) help!!
So sub one equation into the other, which, how ect. Thanks

Answer 1

To answer your question, you are using the pythagorean theorem. a² + b² = c², so you should have x² + y² = 15²

xy = 108
x² + y² = 225

I'd solve it by adding 2xy to both sides:
x² + 2xy + y² = 225 + 2xy

The first part now factors to:
(x + y)(x + y) = 225 + 2xy

And 2xy can be replaced by 2(108) = 216

(x + y)² = 441
(x + y) = sqrt(441)
x + y = 21

Now you have x + y = 21 and xy = 108

Solving for y:
y = 21 - x

Substitute into the second equation:
x(21 - x) = 108

21x - x² = 108

x² - 21x + 108 = 0
(x - 9)(x - 12) = 0

So the dimensions are 9 x 12 (we can't tell which is width and which is length, either way works).

Note, your triangle is 9-12-15, which is triple a 3-4-5 triangle...

Answer 2

15 is also squared. The Pythagorean Theorem says: The SQUARE on the hypotenuse is equal to the sum of the SQUARES on the other two sides.

Draw any right triangle and then draw a square on each side --using the side as one side of the square. The Pythagorean Theorem says that the areas of the two small squares add up to the area of the large square. It is possible to cut the small squares into 5 pieces are reassemble them inside the large one so that they completely cover the large one exactly.

Understand what the Pythagorean Theorem says before you try to use it.