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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:
\[
\boxed{16\sqrt{7}}
\]
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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].?
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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.
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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
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