SOLUTION: The area of the rhombus is 540 cm2; the length of one of its diagonals is 4.5 dm. What is the distance between the point of intersection of the diagonals and the side of the rhombu

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Question 1123955: The area of the rhombus is 540 cm2; the length of one of its diagonals is 4.5 dm. What is the distance between the point of intersection of the diagonals and the side of the rhombus?
Found 3 solutions by greenestamps, MathLover1, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


You will get a lot more out of this problem if you work through it yourself, instead of having me show you all the details. I will describe the steps you can take for one method of finding the answer.

The distance you are looking for is the distance from the intersection of the diagonals to a side of the rhombus.

If you think of finding the area of the rhombus as base times height, then the distance you are looking for is one-half the height of the rhombus.

Then, since you know the area, you can find the length you are looking for if you can find the length of the base of the rhombus -- that is, the length of a side of the rhombus.

The diagonals of a rhombus bisect each other at right angles, forming four right triangles; the legs of each of those triangles are half the lengths of the diagonals; each hypotenuse of one of the triangles is a side of the rhombus.

So....

(1) Use the fact that the area of a rhombus is half the product of the diagonals, along with the given facts that the area is 540 and the length of one diagonal is 45, to find the length of the other diagonal. (Note the length of the diagonal is given as 4.5dm, which is 45cm; you need to use consistent units.)

(2) Use the lengths of the two diagonals to find the lengths of the legs of each of the right triangles, and then use the Pythagorean Theorem to find the length of each hypotenuse.

(3) The hypotenuse of each right triangle is a side of the rhombus, so now you know the length of the "base" of the rhombus.

(4) use the given area of 540 and the length of the base, along with the "base times height" area formula for the rhombus, to find the height of the rhombus. The length you are looking for is half of that height.

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
The area of the rhombus is 540cm%5E2; the length of one of its diagonals is d%5B1%5D=4.5dm=45cm. What is the distance between the point of intersection of the diagonals and the side of the rhombus?

Here is the rhombus with its diagonals.
I put the rhombus inside a rectangular box for safekeeping,
because it looked breakable.
(I know it is not drawn to scale).

You can see that the diagonals split the rhombus into4 congruent right triangles.
Maybe you want to know the distance x (in cm) between the point of intersection of the diagonals and the end of the other diagonal (which is half of the length of the other diagonal).
It is useful to calculate that length, anyway.
The area of a rhombus is the length of one diagonal times half the length of the other.
So, 45%2Ax=540 ---> x=12 .

Maybe you really wanted the distance between the point of intersection of the diagonals and the side of the rhombus, measured along the shortest path, the line perpendicular to the red side.
That is f ,the altitude to the hypotenuse of one of those right triangles.
The length f of such an altitude in a right triangle with leg lengths x and half of the other diagonal which isd%5B1%5D%2F2=45%2F2 can be found from
1%2Ff%5E2=1%2Fx%5E2%2B1%2F%28d%5B1%5D%2F2%29%5E2 .
1%2Ff%5E2=1%2F12%5E2%2B1%2F22.5%5E2
1%2Ff%5E2=1%2F144%2B1%2F506.25
1%2Ff%5E2=%28506.25%2B144%29%2F72900
1%2Ff%5E2=650.25%2F72900
=> f%5E2=72900%2F650.25
f=sqrt%2872900+%2F650.25%29
f=10.588 --> highlight%28f=10.59%29

In other words, the distance from the center of the rhombus to one of its sides is approximately highlight%2810.59cm%29 .


Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
The area of the rhombus is 540cm%5E2; the length of one of its diagonals is d%5B1%5D = 4.5 dm = 45 cm.
What is the distance between the point of intersection of the diagonals and the side of the rhombus?
~~~~~~~~~~~~~~~~~~~~~~~

I will try to produce the solution in more simple form than that by the tutor @MathLover1. 




The area of the rhombus is 4 times the area of the small right-angled triangle formed by its diagonals.


It gives the equation to find "x",  which is half of the second diagonal

4%2A%281%2F2%29%2A%2845%2F2%29%2Ax = 540,   which implies   x = 540%2F45 = 12 cm.



Then the side of the rhombus is equal

sqrt%28%2845%2F2%29%5E2%2B12%5E2%29 = sqrt%28650.25%29 = 25.5 cm.     (Pythagoras)


The area of each of four small right-angled triangle formed by its diagonals is

%281%2F2%29%2Ar%2A25.5 = 540%2F4 cm^2,   


where "r" is the distance under the question,  which gives 


r = %28540%2A2%29%2F%2825.5%2A4%29 = 1080%2F102 = 180%2F17 = 10.59 cm (rounded with 2 decimal places).


Answer.  10.59 cm (rounded with 2 decimal places).

Solved.