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1) E and F are mid points of AB and CD respectively | 1) Given |
2) AE = 1/2 AB and CF = 1/2 CD | 2) Definition of mid point |
3) ABCD is a parallelogram | 3) Given |
4)AB =CD and AB || CD |
4) Properties of parallelogram |
5) AE = FC and AE || FC | 5) From (2) and (4) |
6) AECF is a parallelogram | 6) Properties of parallelogram and from (5) |
7) FA || CE and FQ || CP | 7) Properties of Parallelogram |
8) F is the mid point of CD and FQ ||CP | 8) By Mid Point -Theorem |
9) Q is the mid point of DP ⇒PQ = QD | 9) By Mid Point -Theorem |
10)Similarly, P is the mid point of BQ ⇒ BP = PQ | 10) By mid point theorem |
11) BP= PQ = QD | 11) From (9) and (10) |
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1) PQ || BC | 1) Given |
2) ∠AQP = ∠ABC | 2) Corresponding angles |
3) ∠AQP = 900 | 3) Since ∠ABC =900 |
4) ∠AQP + ∠BQP = 1800 | 4) Linear pair angles |
5) ∠AQP = ∠BQP = 900 | 5) ∠AQP = 90 and from (4) |
6) PQ ⊥ AB | 6) From (5) |
7) P is the mid point of AC and PQ ||BC | 7) Given |
8) Q is the mid point. AQ =BQ | 8) By mid point theorem and definition of mid point. |
9) ∠AQP = ∠BQP | 9) From (5) |
10) PQ =PQ | 10) Reflexive (common) |
11) ΔAPQ = ΔBPQ | 11) SAS Postulate |
12) PA = PB | 12) CPCTC |
13) PA = 1/2 AC | 13) Since P is the mid point of AC |