The parallel sides of a trapezium …

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search-icon-header Search for questions & chapters search-icon-image Class 10 >>Maths >>Triangles >>Basic Proportionality Theorem (Thales Theorem) >>Prove that \"In a trapezium, the line jo Question Bookmark Prove that "In a trapezium, the line joining the mid points of non-parallel sides is (i) parallel to the parallel sides and (ii) Half of the sum of the parallel sides" 564297 expand Easy Solution verified Verified by Toppr Data: In the trapezium ABCD, AD || BC, AX = XB and DY = YC To Prove: (i) XY || AD or XY || BC (ii) XY = 2 1 ​ (AD + BC) Construction: Extend BA and CD to meet at Z. Join A and C. Let it cut XY at P Proof: (i) In △ZBC,AD∣∣BC [∵ Data] ∴ AB ZA ​ = DC ZD ​ [∵ BPT] ∴ 2AX ZA ​ = 2DY ZD ​ [∵ X & Y are mid points of AB and DC] ∴ AX ZA ​ = DY ZD ​ ⇒XY∣∣AD [∵ Converse of B.P.T.] (ii) In △ABC,AX=XB [∵ Data] XP∣∣BC [∵ Proved] ∴AP=PC [∵ Converse of mid point theorem] ∴XP= 2 1 ​ BC [∵ Midpoint Theorem] In △ADC,PY= 2 1 ​ AD By adding, we get XP+PY= 2 1 ​ BC+ 2 1 ​ AD ∴XY= 2 1 ​ (BC+AD)