Two diagonals of a parallelogram have lengths 6 and 10. What is the largest possible length of the shortest side of the parallelogram?
here is a representative image i made:
since both diagonals cot eachother in the middle, they form 4 triangles; by looking at the image, we can say that:
$ \overline{AO}=\frac{6}{2} \ \therefore \ \overline{CO}=\frac{6}{2} $
$ \overline{DO}=\frac{10}{2} \ \therefore \ \overline{BO}=\frac{10}{2} $
now there are two little rules for triangles
1) sum of any two sides $>$ third side.
2) difference of any two sides $<$ third side. ( commonly used only to prove whether the triangle exists or not )
lets consider the $\triangle BOC $ where side $\overline{BC}=$third side (the third and largest side of the triangle, but the smallest side of the parallelogram)
1) $ \frac{10}{2}+\frac{6}{2} > \text{ third side} $
$ 5+3> \text{ third side} $
$ 8> \text{ third side} $
changing the sides, would also change the 'direction' of the less-than-sign, so logically, it would become larger-than-sign:
$ \text{ third side} <8 $