Question 198847
If we just focus on one triangle, we get


{{{drawing(500,500,-0.5,2,-0.5,3.2,
line(0,0,0,3),
line(0,3,2,0),
line(2,0,0,0),
locate(-0.2,1.5,x),
locate(1,-0.2,x),
locate(1,2,6*sqrt(2))
)}}}



To find the unknown length, we need to use the Pythagorean Theorem.



Remember, the Pythagorean Theorem is {{{a^2+b^2=c^2}}} where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.



Since the legs are {{{x}}} and {{{x}}} (ie the legs are equal) this means that {{{a=x}}} and {{{b=x}}}


   

Also, since the hypotenuse is {{{6*sqrt(2)}}}, this means that {{{c=6*sqrt(2)}}}.



{{{a^2+b^2=c^2}}} Start with the Pythagorean theorem.



{{{x^2+x^2=(6*sqrt(2))^2}}} Plug in {{{a=x}}}, {{{b=2}}}, {{{c=6*sqrt(2)}}} 



{{{x^2+x^2=72}}} Square {{{6*sqrt(2)}}} to get {{{(6*sqrt(2))^2=(6*sqrt(2))(6*sqrt(2))=36*(sqrt(2))^2=36*2=72}}}.



{{{2x^2=72}}} Combine like terms.



{{{x^2=72/2}}} Divide both sides by 2.



{{{x^2=36}}} Reduce



{{{x=sqrt(36)}}} Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).



{{{x=6}}} Evaluate the square root of 36 to get 6



So the side length of the square is 6 units.



Now simply plug this side length into the following formulas (note: "s" is the length of the side of the square):



Perimeter: {{{P=4s=4(6)=24}}}. So the perimeter is 24 units.


Area: {{{P=s^2=(6)^2=36}}}. So the area is 36 square units.