Yes, this is a school problem my son brought home. It is not homework but part of a collection of puzzlers.
You have a trapezoid ABCD. The bases (parallel lines) are AB and CD. The two legs (non-parallel lines) are AD and CB.
Leg AD = 10 and base CD = 12. No angles are given.
What is the perimeter of the trapezoid?
Looking for hints to get going.
Is the problem given entirely in words, or is there a diagram? Even if no angles are explicitly given, usually things that look like right angles are allowed to be inferred to be right angles. So if it’s a right trapezoid, this problem becomes a lot easier.
I’m not 100% sure we have enough information if it’s not a right trapezoid.
ETA: An isosceles trapezoid also may be possible.
There is a diagram showing a trapezoid with base DC smaller than base AB. It is not a right trapezoid. There are five possible answers given. The possibilities are:
a) 5*(sqrt(3) + sqrt(2))
b) 41*(sqrt(3) + sqrt(2))
c) 39 + 5*sqrt(5)
d) 39 + 5*(sqrt(3) + sqrt(2))
e) 39 + 10 * (2*sqrt(3) + sqrt(2))
Are you sure that that is the correct question? It seems insoluble.
Draw the trapzoid as ABCD going counter clockwise. Label AD 10 and DC 12 now look at point B and imagine sliding it towards A where at its minimum as it just becomes a triangle (AB=0) AB + BC is between 22 and 12. Now slide it the other way towards infinity. Now AB + BC approaches infinity. All answers between 12 (where the 'trapezoid collapses into a line) and infinity seem possible.
There does not seem to be enough information to decide!
There’s not enough information to solve the problem. Or, put another way, there are infinitely many trapezoids that satisfy the initial conditions.
Even if the trapezoid were isosceles or a right trapezoid there still is not enough information to uniquely solve it.
Yup, even with a right trapezoid you end up with one leg in terms of the unknown base (or vice versa).
edit: Is it possible to work backwards from the multiple choice answers? The solution isn’t unique, but perhaps (without looking at the choices too much), only one satisfies certain necessary constraints?
Given what the choices seem to pivot around, my guess would be that d) was the desired answer [having both the 39 + and and the (sqrt(3) + sqrt(2)) which come up in the others (indeed, even 5*(sqrt(3) + sqrt(2) comes up a lot)]. It’s understandable that then a) indicates forgetting the 39 + term, b) indicates accidentally taking 39 + 5 * x to equal (39 + 5) * x, c) indicates mistakenly assuming sqrt(a) + sqrt(b) to equal sqrt(a + b), and e) is just some random extra factors of 2.
Furthermore, d) is the answer we get if we assume the angle ABC to be 45 degrees and the angle DAB to be 30 degrees. Which are very nice angles trigonometrically that commonly pop up in these problems…
Er, in a world where you also mistakenly take 39 + 5 to equal 41, that is. But who would ever do that? :smack:
Still, I stand by my theory that the desired answer is d) and the missing information is the nice angles.
From the way the question is stated it seems to me like the intentions is to get the student to compute the five expressions more than anything to do with trapezoids. With the conditions given the only thing we know is that the perimeter must be greater than 2*12= 24. Only four of the expressions evaluate to values above 24 and one is under so that one is out.
Thanks for the help everyone. I am having him ask the teacher for clarification.
I will also go along with that. 
The problem is massively underdetermined.
Apparently, this is what the teacher was intending but now realizes there is not enough information as stated to answer the problem.
Thanks for all your help.