SOLUTION: The sum of the lengths of the diagonals of a rhombus of side 61 cm is 142 cm. What is (a) the difference in the lengths of the diagonals, (b) the area of the rhombus? i can't g

Yes, there's enough information.  First we draw a rhombus with 61cm sides:







Now draw in the longer diagonal and label its length x cm.







Now take away the bottom half of the rhombus,

and you have an isosceles triangle. Label the

angle :  







We use the law of cosines:



 





That's one equation we will use.



Now let's go back to the original rhombus.







Since two angles of a parallelogram

which are next to each other are 

supplementary, we label the angle

° 

 

Now we draw the other diagonal and label its length y cm:







Take away the right half and we have 

another isosceles triangle.







We use the law of cosines again:



 





Now we use the fact that 







or







Take that equation with the other one:









Now we add equals to equals.  Adding

the left sides gives  and

adding the right sides, the cosine terms

cancel out.  So we have:







Now we are told that the sum of the diagonals

is 142 cm., so we have the equation







So we have this system of equations:









Can you solve that by solving the second for

one of the letters and substituting into the

first equation?  If not post again asking

how.



Solution to that system of equations: , 



Those are the lengths of the diagonals. 



The difference is just .



Now since the diagonals of a rhombus are 

perpendicular bisectors of each other, the

shorter diagonal, which is 22 cm is bisected 

by the long diagonal, and so each half is

11 cm. 



 



To find the area, take away the bottom half, and

we have this triangle to find the area of:







The base of that triangle is , and its

height is 







So we double that to find the total area of the rhombus.



Answer = 



Edwin