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Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR?
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Sides AB and AC of median AD of triangle ABC are proportional to sides...
Given: Two triangles ΔABC and ΔPQR in which AD and PM are medians such that AB/PQ = AC/PR = AD/PM
To Prove: ΔABC ~ ΔPQR
Construction: Produce AD to E so that AD = DE. Join CE, Similarly produce PM to N such that PM = MN, also Join RN.
Proof: In ΔABD and ΔCDE, we have
AD = DE [By Construction]
BD = DC [∴ AP is the median]
and, ∠ADB = ∠CDE [Vertically opp. angles]
∴ ΔABD ≅ ΔCDE [By SAS criterion of congruence]
⇒ AB = CE [CPCT] ...(i)
Also, in ΔPQM and ΔMNR, we have
PM = MN [By Construction]
QM = MR [∴ PM is the median]
and, ∠PMQ = ∠NMR [Vertically opposite angles]
∴ ΔPQM = ΔMNR [By SAS criterion of congruence]
⇒ PQ = RN [CPCT] ...(ii)
Now, AB/PQ = AC/PR = AD/PM
⇒ CE/RN = AC/PR = AD/PM ...[From (i) and(ii)]
⇒ CE/RN = AC/PR = 2AD/2PM
⇒ CE/RN = AC/PR = AE/PN [∴ 2AD = AE and 2PM = PN]
∴ ΔACE ~ ΔPRN [By SSS similarity criterion]
Therefore, ∠2 = ∠4
Similarly, ∠1 = ∠3
∴ ∠1 + ∠2 = ∠3 + ∠4
⇒ ∠A = ∠P ...(iii)
Now, In ΔABC and ΔPQR, we have
AB/PQ = AC/PR (Given)
∠A = ∠P [From (iii)]
∴ ΔABC ~ ΔPQR [By SAS similarity criterion]
This question is part of UPSC exam. View all Class 10 courses
Most Upvoted Answer
Sides AB and AC of median AD of triangle ABC are proportional to sides...
Proof of Triangle Similarity
- Given Information:
- In triangle ABC, sides AB and AC of median AD are proportional to sides PQ and PR of triangle PQR.
- Median PM of triangle PQR is also proportional to sides AB and AC of triangle ABC.
- To Prove:
- Triangle ABC is similar to triangle PQR.
- Proof:
- Let's denote the lengths of the sides as follows:
- AB = k1 * PQ
- AC = k2 * PR
- PM = k3 * AB
- Since AD is a median of triangle ABC, we know that AD = (AB + AC)/2.
- Substituting the values of AB and AC, we get AD = (k1 * PQ + k2 * PR)/2.
- Similarly, since PM is a median of triangle PQR, PM = (PQ + PR)/2.
- From the given information, we have PM = k3 * AB. Substituting the value of AB, we get PM = k3 * k1 * PQ.
- Conclusion:
- By comparing the ratios of the sides and medians of the two triangles, we can see that the triangles are similar by SSS (Side-Side-Side) similarity criterion.
- Therefore, triangle ABC is similar to triangle PQR.
- Given Information:
- In triangle ABC, sides AB and AC of median AD are proportional to sides PQ and PR of triangle PQR.
- Median PM of triangle PQR is also proportional to sides AB and AC of triangle ABC.
- To Prove:
- Triangle ABC is similar to triangle PQR.
- Proof:
- Let's denote the lengths of the sides as follows:
- AB = k1 * PQ
- AC = k2 * PR
- PM = k3 * AB
- Since AD is a median of triangle ABC, we know that AD = (AB + AC)/2.
- Substituting the values of AB and AC, we get AD = (k1 * PQ + k2 * PR)/2.
- Similarly, since PM is a median of triangle PQR, PM = (PQ + PR)/2.
- From the given information, we have PM = k3 * AB. Substituting the value of AB, we get PM = k3 * k1 * PQ.
- Conclusion:
- By comparing the ratios of the sides and medians of the two triangles, we can see that the triangles are similar by SSS (Side-Side-Side) similarity criterion.
- Therefore, triangle ABC is similar to triangle PQR.
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Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR?
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Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR?.
Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sides AB and AC of median AD of triangle ABC are proportional to sides PQ and PR and median PM of another triangle PQR . show ABC similar to PQR?.
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