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In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square?
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In a triangle of angle aob equal 90 degree if p and q are points of tr...
Introduction
To prove that OP² + OQ² = (5/9) AB² in triangle AOB, where angle AOB is 90 degrees and P and Q are points of trisection of segment AB.
Triangle Setup
- Let O be at the origin (0, 0).
- Point A can be represented as (0, b) and point B as (a, 0).
- The length of AB can be calculated using the distance formula:
AB = √(a² + b²).
Points of Trisection
- Points P and Q divide AB into three equal parts.
- Coordinates of P:
P = (a/3, 2b/3).
- Coordinates of Q:
Q = (2a/3, b/3).
Calculating OP² and OQ²
- OP² = (x_P)² + (y_P)² = (a/3)² + (2b/3)²
- OP² = (a²/9) + (4b²/9) = (a² + 4b²) / 9.
- OQ² = (x_Q)² + (y_Q)² = (2a/3)² + (b/3)²
- OQ² = (4a²/9) + (b²/9) = (4a² + b²) / 9.
Combining OP² and OQ²
- OP² + OQ² = ((a² + 4b²) + (4a² + b²)) / 9
- This simplifies to (5a² + 5b²) / 9 = 5(a² + b²) / 9.
Final Step
- Since AB² = a² + b², we have:
OP² + OQ² = (5/9) AB².
Conclusion
Thus, it is proven that OP² + OQ² = (5/9) AB² in triangle AOB with angle AOB equal to 90 degrees.
To prove that OP² + OQ² = (5/9) AB² in triangle AOB, where angle AOB is 90 degrees and P and Q are points of trisection of segment AB.
Triangle Setup
- Let O be at the origin (0, 0).
- Point A can be represented as (0, b) and point B as (a, 0).
- The length of AB can be calculated using the distance formula:
AB = √(a² + b²).
Points of Trisection
- Points P and Q divide AB into three equal parts.
- Coordinates of P:
P = (a/3, 2b/3).
- Coordinates of Q:
Q = (2a/3, b/3).
Calculating OP² and OQ²
- OP² = (x_P)² + (y_P)² = (a/3)² + (2b/3)²
- OP² = (a²/9) + (4b²/9) = (a² + 4b²) / 9.
- OQ² = (x_Q)² + (y_Q)² = (2a/3)² + (b/3)²
- OQ² = (4a²/9) + (b²/9) = (4a² + b²) / 9.
Combining OP² and OQ²
- OP² + OQ² = ((a² + 4b²) + (4a² + b²)) / 9
- This simplifies to (5a² + 5b²) / 9 = 5(a² + b²) / 9.
Final Step
- Since AB² = a² + b², we have:
OP² + OQ² = (5/9) AB².
Conclusion
Thus, it is proven that OP² + OQ² = (5/9) AB² in triangle AOB with angle AOB equal to 90 degrees.
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In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square?
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In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square?.
In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle of angle aob equal 90 degree if p and q are points of tri section of AB prove that op square + oq square equals 5 by 9 ab square?.
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