SOLUTION: How to solve: Given the area of a rhombus is 24 cm2 and its perimeter is 20 cm., what is its height?

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Question 1030628: How to solve: Given the area of a rhombus is 24 cm2 and
its perimeter is 20 cm., what is its height?
Found 2 solutions by Boreal, Edwin McCravy:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
Draw this.
Area of rhombus is the half the product of the diagonals.
The sides are 5 cm, and each of those is a hypotenuse of the triangles made by the diagonals.
The sides of a right triangle with hypotenuse 5 are 3 and 4.
One diagonal is therefore 6 cm and the other 8 cm. Half their product is 24 cm^2.
The area is bh=24 cm^2
The base is 5 cm, so the height is 4.8 cm.
The area is also a^2*sin A, and the half angle for a corner is tangent (3/4). Double that angle, take the sine of it, and multiply by 25, and the answer is 24.
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!

Here's another way, that doesn't require any special formulas
involving diagonals, just the ordinary formulas for the area of a 
triangle and a rectangle.

Since the perimeter is 20 cm, and all 4 sides are the same length,
each side is 20/4 = 5 cm each.  So we have this figure:



The rhombus is made up of two congruent right triangles
plus a rectangle between them as we can see in the figure 
above.

Since all the sides are 5 cm each we have the equation:



The area of the rhombus is twice the area of one of the
triangles plus the area of the rectangle in the middle.

The area of one of the triangles is 
So twice that is 
The area of the rectangle in the middle is .
Therefore the area of the rhombus, which we set = 24, is




Since 

we can substitute 5 for (x+y) and get:


 or 4.8 cm.

Edwin
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