The rhombus is made of 4 right triangles. The base of each of those
right triangles is x and the height of each one is y.
The area of each of those 4 right triangles is
A = ·b·h
A = ·x·y
So the area of the whole rhombus is 4 times that.
Area of rhombus = 4··x·y = 2xy
We are given that the area of the rhombus is 24 cm², so
2xy = 24
By the Pythagorean theorem on each right triangle:
x² + y² = 5²
x² + y² = 25
So we have this system:
x² + y² = 25
2xy = 24
Add equals to equals:
x² + 2xy + y² = 49
Factor the left side as a perfect square:
(x + Y)² = 49
Use the principle of square roots
x + y = ±7 but we can ignore the negative sign.
x + y = 7
y = 7 - x
Substitute in 2xy = 24
2xy = 24
2x(7-x) = 24
Divide both sides by 2
x(7-x) = 12
7x-x² = 12
-x²+7x-12 = 0
Multiply both sides by -1
x²-7x+12 = 0
Factor the left side
(x-4)(x-3) = 0
x-4 = 0; x-3 = 0
x = 4; x = 3
Substitute each in y = 7 - x
y = 3; y = 4
That looks like two different solutions but actually
they are both the same. One diagonal is 2x = 2(4) = 8 cm
The other diagonal is 2y = 2(3) = 6 cm.
Edwin
