Find the value of the expression $$(800 + 7n - n^2)$$ where n cm is the length of one diagonal of a quadrilateral whose opposite sides are parallel. The adjacent sides of the quadrilateral are 3 cm and 4 cm in length and the other diagonal is 1 cm in length.
Solution
It is said that two opposite sides of the quadrilateral are parallel.
So, it can be Square, Rhombus, Parallelogram, or Rectangle.
It is not square and rhombus because two adjacent sides are unequal(given).
It is not rectangle, because if it is rectangle the diagonal length will be $$\sqrt{\ 3^2+4^2}=5^{ }$$
So, it is a parallelogram.
We know that for a parallelogram $$d_1^2+d_2^2=2\left(a^2+b^2\right)$$.
$$d_1=1$$
a=3
b =4
$$1^2+d_2^2=2\left(3^2+4^2\right)$$.
$$1^2+d_2^2=2\left(3^2+4^2\right)$$.$$1+d_2^2=50\ \Rightarrow d_2^2=49\ \Rightarrow\ d_2=7$$.
The value of the expression $$(800 + 7n - n^2)$$ is $$(800 + 7*7- 7^2)$$=800+49-49=800
