A rectangular painting has a diagonal measure of 13 inches and an area of 60square inches. Use area and Pythagorean Theorem formula to find the length and width

Question

A rectangular painting has a diagonal measure of 13 inches and an area of 60 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

To speed up homework and tests, you should begin to recognize "Pythagorean Triples" that make problems less messy so calculations are easy.  The numbers (5,12,13) are a Pythagorean Triple.  So are: (1,1,√2), (1,√3,2), (3.4.5) ...

 

Here the diagonal is 13 and the sides are 5 and 12 (5*12=60), but lets do the calculations for practice:

 

Let:

   x = length of painting

   y = width of painting

 

Area of painting = 60 = xy

 

Pythagorean Theorem says   x² + y² = 13²

 

    x² + (60/x)² = 13²      [60=xy means y=60/x]

    x⁴ + 3600 = 169x²        [multiply everything by x² to get rid of fraction

        z² + 3600 = 169z        [for now, substitute z for x²]

        z² - 169z + 3600 = 0     [subtract 169z from both sides]

        (z -25)(z -144) = 0

         either z=25 or z=144

       so,

        either x=5 or x=12     (remember x² = z,  we exclude x=-5 and x=-12)

 

Then, recall that xy=60, so

 

Answer: either x=5 and y=12

            or  x=12 and y=5