In the rhombus ABCD, the length of diagonal BD is 6 and the length of diagonal AC is 8. What is the perimeter of ABCD?

This was on my practice GMAT from MGMAT.

Here is what I don't understand

If the Diagonal (1) X Diagonal (2) / 2 = the area and each side is equal to 5, then why is the area not 25.

In other words, why is the area of a rhombus different from the area of a square. If all the sides are equal, then why is the area different?

Thanks,

Square has all 4 internal angles equal to 90 degrees, while a rhombus necessarily doesn't.

So, you would have to calculate the area of one triangle using the 1/2 * base * height formula, and then multiply that result by 4 because a rhombus has 4 identical triangles.

1) a rhombus is a quadrilateral whose four sides are all congruent

2) a square is a quadrilateral whose four sides are all congruent
and whose angles are all right angles

=> a square is a rhombus that is also a rectangle

These are both rhombuses:

+--------+
| | +--------+
| | / /
| | / /
| | / /
+--------+ +--------+


Square is a special case of rhombus. You should be able not use formula applicable to
Rhombus on a square but not otherway. You cant use area= side ^2 for any rhombus except for rhombus which is a square.

The image in my above reply has got distorted
One is a rhombus with all angles 90 degrees and other is a rhombus with angle 45degrees[/img]

just think about it like this: a rhombus is the diamond from a deck of playing cards. it has tons of symmetry, and is much, much easier to visualize accurately than, say, a parallelogram or kite.

all of the trouble people have with rhombuses (actually 'rhombi') stems directly from their trying to draw the figures lying on one side, rather than in the easy, vertically-and-horizontally-symmetrical configuration of the diamond.

you can't find the area of the diamond by multiplying the sides together, as you can 'flex' the diamond to change its area without affecting the lengths of the sides. so, if all the sides are 5, you can still make the area extremely close to 0 (by making the diamond extremely long and skinny), or you can make the area as large as 25 (by making the diamond a square).

yeah, the forum auto-formats your text. sorry about that :S

That's really helpful to think of a rhombus as a diamond that can be compressed to 0 and expanded to 25. Thanks a bunch!

I like it too!