UPSC Exam > UPSC Questions > 4. In triangle ABC, let O be the circumcenter...
Start Learning for Free
4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in
BC, CA, AB respectively.
Let BC = 12, AC = 10, AB = 6. Find [P QR].?
BC, CA, AB respectively.
Let BC = 12, AC = 10, AB = 6. Find [P QR].?
Most Upvoted Answer
4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the refl...
Understanding the Triangle and Its Elements
To solve for the area of triangle \( PQR \) formed by the reflections of the circumcenter \( O \) of triangle \( ABC \) across its sides, we first need to identify the properties and measurements of triangle \( ABC \):
- Sides of Triangle ABC:
- \( BC = 12 \)
- \( AC = 10 \)
- \( AB = 6 \)
Calculating the Area of Triangle ABC
We can use Heron's formula to determine the area of triangle \( ABC \).
- Semi-perimeter \( s \):
\[
s = \frac{AB + AC + BC}{2} = \frac{6 + 10 + 12}{2} = 14
\]
- Area \( [ABC] \):
\[
[ABC] = \sqrt{s(s - AB)(s - AC)(s - BC)} = \sqrt{14 \times (14 - 6) \times (14 - 10) \times (14 - 12)}
\]
\[
= \sqrt{14 \times 8 \times 4 \times 2} = \sqrt{448} = 8\sqrt{7}
\]
Finding the Area of Triangle PQR
The area of triangle \( PQR \) can be derived from the properties of the reflection of the circumcenter:
- The area of triangle \( PQR \) is equal to the area of triangle \( ABC \), given that reflections maintain the area.
- Therefore, \( [PQR] = 2 \times [ABC] \).
- Thus,
\[
[PQR] = 2 \times 8\sqrt{7} = 16\sqrt{7}
\]
Conclusion
The area of triangle \( PQR \) is:
\[
\boxed{16\sqrt{7}}
\]
To solve for the area of triangle \( PQR \) formed by the reflections of the circumcenter \( O \) of triangle \( ABC \) across its sides, we first need to identify the properties and measurements of triangle \( ABC \):
- Sides of Triangle ABC:
- \( BC = 12 \)
- \( AC = 10 \)
- \( AB = 6 \)
Calculating the Area of Triangle ABC
We can use Heron's formula to determine the area of triangle \( ABC \).
- Semi-perimeter \( s \):
\[
s = \frac{AB + AC + BC}{2} = \frac{6 + 10 + 12}{2} = 14
\]
- Area \( [ABC] \):
\[
[ABC] = \sqrt{s(s - AB)(s - AC)(s - BC)} = \sqrt{14 \times (14 - 6) \times (14 - 10) \times (14 - 12)}
\]
\[
= \sqrt{14 \times 8 \times 4 \times 2} = \sqrt{448} = 8\sqrt{7}
\]
Finding the Area of Triangle PQR
The area of triangle \( PQR \) can be derived from the properties of the reflection of the circumcenter:
- The area of triangle \( PQR \) is equal to the area of triangle \( ABC \), given that reflections maintain the area.
- Therefore, \( [PQR] = 2 \times [ABC] \).
- Thus,
\[
[PQR] = 2 \times 8\sqrt{7} = 16\sqrt{7}
\]
Conclusion
The area of triangle \( PQR \) is:
\[
\boxed{16\sqrt{7}}
\]
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].?
Question Description
4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].?.
4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].?.
Solutions for 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? defined & explained in the simplest way possible. Besides giving the explanation of
4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].?, a detailed solution for 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? has been provided alongside types of 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? theory, EduRev gives you an
ample number of questions to practice 4. In triangle ABC, let O be the circumcenter. Let P, Q, R be the reflection of O in BC, CA, AB respectively. Let BC = 12, AC = 10, AB = 6. Find [P QR].? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup