Question

the length of one side of a rhombus is 41 cm and its area is 720 CM square what is the sum of the length of its diagonal calculate

Answer 1

Given that ABCD is a rhombus means all sides are equal in length

Given that side length of rhombus = 41 cm

so

AB=BC=CD=DE=41

Given that area of rhombus = 720 square cm

We know that area of rhombus is given by formula

Area = Altitude * side

720 = DE * 41

720/41 = DE

17.56=DE

We see that CDE is a right angle triangle so we can use Pythogorean theorem to find value of CE

$CE^2+DE^2=DC^2$

$CE^2+17.56^2=41^2$

$CE^2=41^2-17.56^2$

$CE^2=1372.6464$

CE=37.0492429072

which is approx

CE=37.05

According to diagram,

BE=BC+CE = 41+37.05 = 78.05

BE=78.05

Now apply Pythogorean theorem to find value of BD in right angle triangle BED

$BD^2=BE^2+DE^2$

$BD^2=78.05^2+17.56^2$

$BD^2=6400.1561$

BD=80.0009756191

Which is approx BD=80

We also know that area of rhombus is given by formula

$Area=\frac{1}{2}(Diagonal_1 + Diagonal_2 )$

$Area=\frac{1}{2}(BD + AC )$

$720=\frac{1}{2}(80 + AC )$

720*2=80 + AC

1440=80 + AC

1360=AC

We need to find the sum of the lengths of it's diagonals so we add AC and BD

Sum = AC+BD= 1360+80 = 1440

Hence final answer is 1440 cm.