A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set 5,6,7,8,9,10

Step-by-step solution

Without losing generality, assume a>c and d>b and sides A B | C D aligned & d^2-x^2=b^2-(c-a+x)^2, Similarly d^2-x^2=b^2-(a-c-x)^2 & x= (a-c)^2+d^2-b^2 2(a-c) x= (a-c)^2+d^2-b^2 2(a-c) aligned If x (0, d) , then there will be unique trapezoid aligned & (a-c)^2+d^2-b^2 2 c(c-a) (0, d) & (a-c)^2+d^2-b^2-2 d(a-c) 0 a+b>c+d aligned And a>c, d>b Using these inequality, numerate these pairs ( a, b, c, d ) Case I : a=10 total no. of cases =16 Case II : a=9 total no. of cases =9 Case III : a=8 total no. of cases =4 Case (I