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

Made by

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 and Computer Science at Teachoo

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