Math Question

Question

A rectangular painting has a diagonal measure of 25 inches and an area of 168 square inches. Use the formula for the area of a rectangle and the Pythagorean Theorem to find the length and width of the painting.

Answer 1

Let length of rectangle = x

Let width of rectangle = y

 

Using these variables, we can create equation that uses the area of triangle and Pythagorean theorem.

 

x² + y² = 25²

x² + y² = 625           eq1   -----> Pythagorean theorem

 

xy = 168                  eq2   -----> area of rectangle

 

 

We can now substitute eq1 into eq2 to get eq2 in terms of only x.

 

x√(625 - x²) = 168

 

 

Square both sides of the equation to undo the square-root.

 

x²(625 - x²) = 28224

 

625x² - x⁴ = 28224

 

 

Move all terms to the right side of equation.

 

0 = x⁴ - 625x² - 28224

 

 

Let      q = x²

 

Then,

 

q² - 625q - 28224 = 0

 

 

Use the quadratic formula to solve for q.  Select the positive solution of q.  Then once you have q, take the square-root of the result to get x.  This is your length.  Once you have x, use the formula

 

y = 168 / x

 

to get your width.