2020 · Q100
Consider the following statements with reference to the given figure:
- 1The sum of the areas of ΔAOD and ΔBOC is equal to the sum of the areas of ΔAOB and ΔDOC.
- 2∠AOD = ∠BOC
- 3AB + BC + CD + DA > AC + BD
Which of the above statements are correct?
Elementary MathematicsQuadrilaterals and Polygons
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- D1, 2 and 3
Explanation
Statement 2: ∠AOD and ∠BOC are vertical angles, hence equal. Statement 3: applying the triangle inequality in △ABC, △CDA, △BCD and △DAB and adding gives 2(AB + BC + CD + DA) > 2(AC + BD), i.e. the sum of the sides exceeds the sum of the diagonals. Statement 1 fails: with AD ∥ BC but AD ≠ BC, area(△AOD) + area(△BOC) = ½(AD·h₁ + BC·h₂) while area(△AOB) + area(△DOC) = BC·h₁, which are equal only if AD = BC (a parallelogram). So only statements 2 and 3 are correct.