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PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)?
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Proof:
Given: PQRS is a quadrilateral with P as the center of the circle passing through Q, R, and S.
To prove: ∠QPS = 2(∠RQS + ∠RSQ)
Construction:
1. Draw the circle with center P passing through Q, R, and S.
2. Join PQ, PR, and PS to form the quadrilateral PQRS.
Proof:
Step 1: ∠PQR = ∠PSR
- Angles subtended by the same arc QS are equal (angle at the center is twice the angle at the circumference).
- Therefore, ∠PQS = ∠PRS
Step 2: ∠PQS = ∠RQS + ∠RSQ
- In triangle PQS, the sum of the angles is 180 degrees.
- ∠PQS + ∠QPS + ∠PSQ = 180 degrees
- ∠PSQ = 90 degrees (angle in a semicircle)
- Substituting the values, we get:
∠PQS + ∠QPS + 90 degrees = 180 degrees
∠PQS + ∠QPS = 90 degrees
- ∠PQS = ∠RQS + ∠RSQ (from Step 1)
- Substituting the values, we get:
∠RQS + ∠RSQ + ∠QPS = 90 degrees
Step 3: ∠QPS = 2(∠RQS + ∠RSQ)
- Rearranging the equation from Step 2, we get:
∠QPS = 90 degrees - (∠RQS + ∠RSQ)
∠QPS = 90 degrees - ∠QPS
2∠QPS = 90 degrees
∠QPS = 45 degrees
- ∠RQS + ∠RSQ = 45 degrees (from Step 2)
- Multiplying by 2 on both sides, we get:
2(∠RQS + ∠RSQ) = 2(45 degrees)
2(∠RQS + ∠RSQ) = 90 degrees
Therefore, ∠QPS = 2(∠RQS + ∠RSQ), which proves the given statement.
Given: PQRS is a quadrilateral with P as the center of the circle passing through Q, R, and S.
To prove: ∠QPS = 2(∠RQS + ∠RSQ)
Construction:
1. Draw the circle with center P passing through Q, R, and S.
2. Join PQ, PR, and PS to form the quadrilateral PQRS.
Proof:
Step 1: ∠PQR = ∠PSR
- Angles subtended by the same arc QS are equal (angle at the center is twice the angle at the circumference).
- Therefore, ∠PQS = ∠PRS
Step 2: ∠PQS = ∠RQS + ∠RSQ
- In triangle PQS, the sum of the angles is 180 degrees.
- ∠PQS + ∠QPS + ∠PSQ = 180 degrees
- ∠PSQ = 90 degrees (angle in a semicircle)
- Substituting the values, we get:
∠PQS + ∠QPS + 90 degrees = 180 degrees
∠PQS + ∠QPS = 90 degrees
- ∠PQS = ∠RQS + ∠RSQ (from Step 1)
- Substituting the values, we get:
∠RQS + ∠RSQ + ∠QPS = 90 degrees
Step 3: ∠QPS = 2(∠RQS + ∠RSQ)
- Rearranging the equation from Step 2, we get:
∠QPS = 90 degrees - (∠RQS + ∠RSQ)
∠QPS = 90 degrees - ∠QPS
2∠QPS = 90 degrees
∠QPS = 45 degrees
- ∠RQS + ∠RSQ = 45 degrees (from Step 2)
- Multiplying by 2 on both sides, we get:
2(∠RQS + ∠RSQ) = 2(45 degrees)
2(∠RQS + ∠RSQ) = 90 degrees
Therefore, ∠QPS = 2(∠RQS + ∠RSQ), which proves the given statement.
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PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)?
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PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)?.
PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PQRS is such a quadrilateral that P is the centre of the circle which is passing through Q, R and S. Then prove that angle QPS=2(angleRQS + angle RSQ)?.
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