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4. In 📐 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 📐 ABC, let O be the circumcenter. Let P, Q, R be the reflection...
Understanding the Problem
To find the area of triangle PQR, where P, Q, and R are the reflections of the circumcenter O of triangle ABC across the sides BC, CA, and AB respectively, we need to establish some geometric properties and calculations.
Given Dimensions
- BC = 12
- AC = 10
- AB = 6
Finding the Area of Triangle ABC
- We can use Heron's Formula to find the area of triangle ABC.
- Semi-perimeter \( s = \frac{BC + AC + AB}{2} = \frac{12 + 10 + 6}{2} = 14 \)
- Area \( [ABC] = \sqrt{s(s - a)(s - b)(s - c)} \)
- Here, \( a = BC = 12, b = AC = 10, c = AB = 6 \)
- Area calculation:
\[
[ABC] = \sqrt{14(14 - 12)(14 - 10)(14 - 6)} = \sqrt{14 \times 2 \times 4 \times 8} = \sqrt{448} = 8\sqrt{7}
\]
Finding Area of Triangle PQR
- The area of triangle PQR is twice the area of triangle ABC due to the properties of reflections in the circumcenter.
\[
[PQR] = 2 \times [ABC] = 2 \times 8\sqrt{7} = 16\sqrt{7}
\]
Final Answer
The area of triangle PQR is \( \boxed{16\sqrt{7}} \).
To find the area of triangle PQR, where P, Q, and R are the reflections of the circumcenter O of triangle ABC across the sides BC, CA, and AB respectively, we need to establish some geometric properties and calculations.
Given Dimensions
- BC = 12
- AC = 10
- AB = 6
Finding the Area of Triangle ABC
- We can use Heron's Formula to find the area of triangle ABC.
- Semi-perimeter \( s = \frac{BC + AC + AB}{2} = \frac{12 + 10 + 6}{2} = 14 \)
- Area \( [ABC] = \sqrt{s(s - a)(s - b)(s - c)} \)
- Here, \( a = BC = 12, b = AC = 10, c = AB = 6 \)
- Area calculation:
\[
[ABC] = \sqrt{14(14 - 12)(14 - 10)(14 - 6)} = \sqrt{14 \times 2 \times 4 \times 8} = \sqrt{448} = 8\sqrt{7}
\]
Finding Area of Triangle PQR
- The area of triangle PQR is twice the area of triangle ABC due to the properties of reflections in the circumcenter.
\[
[PQR] = 2 \times [ABC] = 2 \times 8\sqrt{7} = 16\sqrt{7}
\]
Final Answer
The area of triangle PQR is \( \boxed{16\sqrt{7}} \).
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4. In 📐 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 📐 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 📐 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 📐 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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