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Example 6 - ABCD is a parallelogram in which P and Q - Examples

Example 6 - Chapter 8 Class 9 Quadrilaterals - Part 2

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Example 6 ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R, show that: APCQ is a parallelogram. Given: ABCD is a parallelogram P , Q are mid-points of AB & CD respectively To prove: APCQ is a parallelogram Proof: Since AB ∥ DC ⇒ AP ∥ QC Also, AB = CD 1/2 AB = 1/2 CD AP = QC Since AP ∥ QC & AP = QC One pair of opposite sides of APCQ are equal and parallel ∴ APCQ is a parallelogram Example 6 ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD.If AQ intersects DP at S and BQ intersects CP at R, show that: (ii) DPBQ is a parallelogram. Since AB ∥ DC ⇒ PB ∥ DQ Also, AB = CD ∴ 1/2 AB = 1/2 CD PB = DQ Now, PB ∥ DQ & PB = DQ Since one pair of opposite sides of DPBQ are equal and parallel ∴ DPBQ is a parallelogram Example 6 ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD.If AQ intersects DP at S and BQ intersects CP at R, show that: (iii) PSQR is a parallelogram. In part(ii) we proved DPBQ is a parallelogram So, DP ∥ QB ∴ SP ∥ QR Also, in part(i) we proved APCQ is a parallelogram So, AQ ∥ PC ∴ SQ ∥ PR From (1) & (2) SP ∥ QR & SQ ∥ PR Now, in PSQR both pairs of opposite sides of PSQR are parallel ∴ PSQR is a parallelogram

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.