2012 · Q54
Let ABCD be a parallelogram. Let m, n be positive integers such that n < m < 2n. Let AC = 2mn and BD = m^2 - n^2. Let AB = (m^2 + n^2)/2.
Statement IAC > BD.
Statement IIABCD is a rhombus.
Which one of the following is correct in respect of the above statements?
Elementary MathematicsGeometry
- ABoth statement-I and statement-II are true and statement-II is the correct explanation of statement-I
- BBoth statement-I and statement-II are true but statement-II is not the correct explanation of statement-I
- CStatement-I is true, but statement-II is false
- DStatement-II is true, but statement-I is false
Explanation
AC-BD = 2mn-(m^2-n^2) = 2n^2-(m-n)^2 > 0 since m < 2n, and the parallelogram law gives AD = AB so it is a rhombus; but being a rhombus does not explain AC > BD.