SOLUTION: If the length of each side of a rhombus is √90 . Find the diagonals in the rhombus ?

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Question 1117324: If the length of each side of a rhombus is √90 . Find the diagonals in the rhombus ?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you don't have enough information to find the length of the diagonals.

you need to know the angles formed by the rhombus.

once you know those, you can find the length of each diagonal.

the opposite angles of your rhombus are congruent.

the angles adjacent to these are supplementary to them and also congruent to each other.

here are the properties of a rhombus.

http://www.dummies.com/education/math/geometry/properties-of-rhombuses-rectangles-and-squares/

note that a rhombus is a special type of parallelogram.

here are the properties of a parallelogram.

http://www.dummies.com/education/math/geometry/properties-of-parallelograms/

if all you know is the length of the sides of the rhombus, then you need to know the angles formed by the thombus as well.

once you know that, you can use trigonometry to find the length of each diagonal.

since the angles formed by the can be an infinite number of degrees between the angles of 0 and 180 degrees, then the length of the diagonals of the rhombus can be an infinite number of lengths determined by those degrees.

once you know one of the angles of the rhombus, you can solve for the length of both diagonals of the rhombus.

the formula you can use is:

length of one of the diagonals of the rhombus is 2 * sin(T/2) * length of side of rhombus.

length of the other of the diagonals of the rhombus is 2 * cos(T/2) * length of side of rhombus.

T is one of the angles of the rhombus.

the rhombus will have 4 angles.

2 of those angles will be congruent to each other (opposite angles are congruent).

2 of those angles will be supplementary to the angles that are congruent to each other, making those other 2 angles also congruent to each other.

you can use any one of these 4 angles to find the length of each diagonal.

for example:

assume your angle is 30 degrees.

then the adjacent angle will be 180 - 30 = 150 degrees.

the length of one of the diagonals of the rhombus will be 2 * sin(30/2) * sqrt(90) = 4.910746106.

the length of the other diagonal of the rhombus will be 2 * cos(30/2) * sqrt(90) = 18.32715397.

if you had chosen the adjacent angle of 150 degrees, you would have gotten the same answer.

the length of one of the diagonals of the rhombus will be 2 * sin(150/2) * sqrt(90) = 18.32715397.

the length of the other diagonal of the rhombus will be 2 * cos(150/2) * sqrt(90) = 4.910746106.

bottom line:

you don't have enough information to find the length of each diagonal if all that you have is the length of the side of the rhombus.

here's the diagram of my rhombus that has side length of sqrt(90) and angles of 30 and 150.

$$$
note that a different angle will lead to different lengths of the diagonals.

that's why just having the length of a side of the rhombus, and nothing else, is not sufficient.