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CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ?
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CM and RN are respectively the medians of triangle ABC and triangle PQ...
Proof of Similarity and Proportions in Triangles
Similarity of Triangles
To prove that triangle AMC~triangle PNR, we need to show that they have the same shape.
Proportions in Triangles
To prove the second and third statements, we need to use the properties of medians.
Therefore, we have proved all the three statements.
Similarity of Triangles
To prove that triangle AMC~triangle PNR, we need to show that they have the same shape.
- Since CM and RN are medians, they divide their respective triangles into two equal halves.
- Therefore, we can say that AM = MB and PN = NQ.
- Also, we know that triangle ABC~triangle PQR.
- Therefore, we have AB/PQ = BC/QR = AC/PR.
- Now, we can say that AM/PN = AB/PQ and BM/QN = BC/QR.
- Substituting the values we know, we get AM/PN = BM/QN = AC/PR.
- Therefore, we can conclude that triangle AMC~triangle PNR.
Proportions in Triangles
To prove the second and third statements, we need to use the properties of medians.
- Let's consider the triangle ABC and its medians CM and BN.
- By the properties of medians, we know that CM = 1/2AB and BN = 1/2AC.
- Similarly, for triangle PQR and its medians RN and QM, we have RN = 1/2PQ and QM = 1/2PR.
- Now, we need to prove that CM/RN = AB/PQ and CMB~RNQ.
- Using the values we know, we get CM/RN = (1/2AB)/(1/2PQ) = AB/PQ.
- Therefore, we have proved the second statement.
- For the third statement, we can use the fact that medians divide a triangle into six smaller triangles of equal area.
- Therefore, we have two sets of three triangles each, which are similar to each other.
- These sets are triangle CMB, triangle AMN, and triangle ACB, and triangle RNQ, triangle QNP, and triangle PQR.
- Therefore, we can conclude that triangle CMB~triangle RNQ.
Therefore, we have proved all the three statements.
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CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ?
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CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ?.
CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for CM and RN are respectively the medians of triangle ABC and triangle PQR .If triangle ABC ~triangle PQR ,prove that i. triangle AMC~triangle PNR ii CM/RN =AB/PQ iii triangle CMB~ triangle RNQ?.
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