Find area of rectangle when given perimeter and diagonal

Question

I was helping my niece (going into 10th) with summer math, and am stumped.

Given: A rectangle, perimeter = 70; diagonal = 25. Find the area.

I am sure this is not supposed to be some weird calculus problem ;-)

she only has finished 9th grade. Can you give me a direction?

Answer 1

perimeter = 2(length + width)

70 = 2(L+ w)

L + w = 35

 

The diagonal divides the rectangle into

2 right triangles where the diagonal is

the hypotenuse of each.  So we have a

right triangle with the hypotenuse of 25 

and the sum of the legs is 35.

If one leg is a, the other is 35-a

 

From pythagorean theorem a² + b² = c² ....

 

a² + (35-a)² = 25²

a² + 1225 - 70a + a² = 625

2a² - 70a + 1225 - 625 = 0

2a² - 70a + 600 = 0

we can factor out a 2

a² - 35a + 300 = 0

 

Ths can be factored by grouping:

Factors of 300 that add to -35 are (-15)(-20).

Replace -35a with -20a - 15a

 

a² - 20a - 15a + 300 = 0

a(a - 20) -15(a - 20) = 0

(a - 15)(a - 20) = 0

 

Either a = 15 or a = 20

 

The rectangle has length 20, width 15

 

Area = length*width = 20(15) = 300 units²