Perimeter of Quadrilateral

Question

The lengths of two sides of a quadrilateral are equal to 1 and 4. One of the diagonals has a lengths of 2 and divide the quadrilateral into two isosceles triangles. What is the perimeter of the quadrilateral?

Answer 1

The length of the diagonal 2 and a side 1 means that the side 4 does not share the same triangle with the side 1. This means that you have the two isosceles triangle with sides: (1,1) and (4,4). The perimeter of the triangle is 10.

Answer 2

I think you are making this very complicated ...first thing is that you were indeed going in the right direction but see the figure $1$ that you have made closely..

Note that the triangle ∆BCD has sides $1,1,2$ so this means-sum of length of the sides $BC,CD$ that is $2$ is equal to $BD$ that is $2$ also but if you know about triangular inequality you must have realised by now that such a triangle cannot be made (because according to triangular inequality if the sum of two sides is equal to the third one then it is actually not a triangle)

So that means the First quadrilateral you have made is not possible.so what should we now,?? Well,we should go in a systematic way instead of just shooting arrows in the air.

Step 1: Make a rough figure(such as you have made)First fix the sides that you know(that is just using the question and leaving the sides that are not known) so we have now made a quadrilateral $ABCD$ in which $AD=4$,$DC=1$,$BD=2$ are fixed.

Step 2: Think about the various possibilities. Note that the question says that the triangle $BCD$ must be isosceles but we have already shown that BC cannot be $1$(by triangular inequality) so that leaves us with only one possibility that is $BC=2$.So half of the question is done.

Now only one side $AB$ is not known.OK...but we know that $∆BAD$ is isosceles so we have again got two possibilities that is $AB=4$ or $AB=2$ but then again if $AB=2$ ..then by the same reasoning as above this is also not possible(by triangular inequality) so that means $AB=4$ is the only possibility left.

Thus now we have got all the sides $AD=4$,$DC=1$,$AB=4$,$BC=2$ adding which we get answer $11$.

Is this information enough.if you need more explainiation just drop a comment below.