2020 · Q79
A line through the vertex A of a parallelogram ABCD meets DC in P and BC produced in Q. If P is the mid-point of DC, then which of the following is/are correct?
- 1Area of ΔPDA is equal to that of ΔPCQ
- 2Area of ΔQAB is equal to twice that of ΔPCQ
Select the correct answer using the code given below:
Elementary MathematicsQuadrilaterals and Polygons
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Since P is the midpoint of DC, △PDA ≡ △PCQ by ASA (PD = PC, alternate interior angles at D and C, vertical angles at P) — so statement 1 holds. From the congruence CQ = AD = BC, so C is the midpoint of BQ; with AB ∥ PC, △QAB ~ △QPC in ratio 2 : 1, hence area ratio 4 : 1, not 2 : 1 — statement 2 fails. Hence only statement 1 is correct.