Examples

Chapter 6 Class 10 Triangles
Serial order wise

### Transcript

Example 8 In Fig. 6.33, CM and RN are respectively the medians of Δ ABC and Δ PQR. If Δ ABC ~ Δ PQR, prove that : (i) Δ AMC ~ Δ PNR Given: Δ ABC and Δ PQR CM is the median of Δ ABC and RN is the median of Δ PQR Also , Δ ABC ~ Δ PQR To Prove: Δ AMC ~ Δ PNR Proof: CM is median of Δ ABC So, AM = MB = 1/2 AB Similarly, RN is the median of Δ PQR So, PN = QN = 1/2 PQ Given ∆ 𝐴𝐵𝐶 ~ ∆ 𝑃𝑄𝑅 𝐴𝐵/𝑃𝑄=𝐵𝐶/𝑄𝑅=𝐶𝐴/𝑅𝑃 𝐴𝐵/𝑃𝑄=𝐶𝐴/𝑅𝑃 (2 𝐴𝑀)/(2 𝑃𝑁)=𝐶𝐴/𝑅𝑃 𝐴𝑀/𝑃𝑁=𝐶𝐴/𝑅𝑃 Also, since ∆ 𝐴𝐵𝐶 ~ ∆ 𝑃𝑄𝑅 ∠ A = ∠ P In Δ AMC & ΔPNR ∠ A = ∠ P 𝐴𝑀/𝑃𝑁 " = " 𝐶𝐴/𝑅𝑃 Hence by SAS similarly ΔAMC ∼ ΔPNR Hence proved Example 8 In Fig. 6.33, CM and RN are respectively the medians of Δ ABC and Δ PQR. If Δ ABC ~ Δ PQR, prove that : (ii) 𝐶𝑀/𝑅𝑁=𝐴𝐵/𝑃𝑄 In part (i) we proved that Δ AMC ~ Δ PNR So, 𝐶𝑀/𝑅𝑁=𝐴𝐶/𝑃𝑅= 𝐴𝑀/𝑃𝑁 Therefore, 𝐶𝑀/𝑅𝑁= 𝐴𝑀/𝑃𝑁 𝐶𝑀/𝑅𝑁= 2𝐴𝑀/2𝑃𝑁 𝐶𝑀/𝑅𝑁=𝐴𝐵/𝑃𝑄 Hence Proved Example 8 In Fig. 6.33, CM and RN are respectively the medians of Δ ABC and Δ PQR. If Δ ABC ~ Δ PQR, prove that : (iii) Δ CMB ~ Δ RNQ Given ∆ 𝐴𝐵𝐶 ~ ∆ 𝑃𝑄𝑅 𝐴𝐵/𝑃𝑄=𝐵𝐶/𝑄𝑅=𝐶𝐴/𝑅𝑃 𝐴𝐵/𝑃𝑄=𝐵𝐶/𝑄𝑅 (2 𝐵𝑀)/(2 𝑄𝑁)=𝐵𝐶/𝑄𝑅 𝐵𝑀/𝑄𝑁=𝐵𝐶/𝑄𝑅 Also, since ∆ 𝐴𝐵𝐶 ~ ∆ 𝑃𝑄𝑅 ∠ B = ∠ Q Now in Δ CMB & ΔRNQ ∠𝐵=∠𝑄 𝐵𝑀/𝑄𝑁=𝐵𝐶/𝑄𝑅 Hence by SAS similarly ΔCMB ∼ ΔRNQ Hence proved

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.