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14. Sides AB and AC and median AD of atriangle ABC are respectivelyproportional to sides PQ and PR andmedian PM of another triangle PQR.Show that AABC-APOR. |
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Answer» Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM ofΔ PQR (see Fig. 6.41). Show thatΔ ABC ~ Δ PQR. Answer:To Prove:Δ ABC∼Δ PQRGiven: Proof:Median divides the opposite sideBD =and, QM = Now,Multiplying and dividing by 2, we get,=  In Δ ABD and Δ PQM, Side-Side-Side (SSS) Similarity Theorem - If the lengths of thecorrespondingsides of two triangles areproportional, then the triangles must besimilar.ΔABDΔPQM (By SSS similarity) ∠ABD =∠PQM (Corresponding angles of similar triangles) In ΔABC and ΔPQR, ∠ABD =∠PQM (Proved above) The SAS Similarity Theorem states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both arecongruent, then the two triangles are similar.ΔABCΔPQR (By SAS similarity)Hence, Proved. |
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