Short vector question help please

ABCD is a rectangle in which AC= a and BD= b, express

AB.AD in terms of |a| and |b| ans ¼( |a|²- |b|² )

hence show that the diagonals are equal in length

any help would be appreciated but please show working

Let diagonals meet at X Then AX = a/2 and BX = b/2

So AB = AX + XB = a/2 - b/2 and AD = AX + XD = a/2 + b/2.

So AB. AD = (a/2 - b/2).(a/2 + b/2) = ¼( |a|²- |b|² )

Since ABCD is a rectangle AB .AD = 0 so ¼( |a|²- |b|² ) = 0 and so |a| = |b| and so the diagonals are

equal in length .