A nice little parallelogram problem

Geometry Level 4

A given parallelogram has sides measuring $7$ and $9$ , and both its diagonals have integer lengths.

Find the sum of all possible products of the lengths of the diagonals. (i.e if the lengths of the diagonals could be $(a_1,b_1)$ , $(a_2,b_2)$ , $\dots$ , $(a_n,b_n)$ , with $a_k\leq b_k$ , then insert you answer as $a_1 b_1+a_2 b_2+\dots + a_n b_n$ .

The answer is 112.

1 solution

X X
Aug 24, 2018

Let the diagonals be $a$ and $b$ . If it's angles are $\theta$ and $180^\circ-\theta$ , then $a^2=7^2+9^2-2\times7\times9\times\cos\theta,b^2=7^2+9^2+2\times7\times9\times\cos\theta$

Solve for $a^2+b^2=260$ and discover the possible answers are $(8,14)$ and $(2,16)$ . $(2,16)$ doesn't fit because it occurs at $\theta=0^\circ$ or $180^\circ$

Hence, the answer is $8\times14=112$

@X X @Freddie Hand I solved it in the same way. But why don't we count the solution twice? I mean, at first I supposed $(a, b) \neq (b, a)$

Jacopo Piccione - 2 years, 9 months ago

Because the result shows that the two parallelogram are congruent, so don't count it twice.

X X - 2 years, 9 months ago

Yes, maybe notation in Freddie's last sentence confused me

Jacopo Piccione - 2 years, 9 months ago

@Jacopo Piccione – Maybe @Freddie Hand should add $a_k\le b_k$

X X - 2 years, 9 months ago

@X X – Ok I've added that, thanks!

Freddie Hand - 2 years, 9 months ago

A nice little parallelogram problem

The answer is 112.

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