Question

The area of a rhombus is 84 sqm.,if its perimeter is 40 m. ,then find its altitude.

Answer 1

each side is 10 units
let the height be h

area of rhombus = base x height
10 x height = 84
height = 84/10 = 8.4

That's it.
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You are not going to believe how I "overthought" this question.
Here is what I did originally:


2 main properties we can use:
a rhomus has 4 equal sides
the diagonals of a rhomus right-bisect each other

let the diagonals have length 2a and 2b
the 4 triangles created are equal
so each triangle has area 21

so (1/2)ab = 21
ab = 42 ----> b = 42/a

also in each triangle, the hypotenuse is 10
a^2 + b^2 = 100
a^2 + 1764/a^2 = 100
a^4 - 100a^2 + 1764 = 0
I was expecting this to factor, but ....

a^2 = (100 ± √2944)/2
= 77.129... or 22.87...

a = √77.129 ... = 8.7823..
or
a = √22.87... = 4.7823..

if a = 8.7823.. , then b = 4.7823
if a = 4.7823.. , then b = 8.7823..

looks like we have "symmetrical" answers
and the long diagonal is 2(8.7823) = 17.56
the shorter diagonal is 2(4.7823) = 9.56

Answer 2

8.4

Answer 3

9m