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The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1?
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The given figure ,o is the centre of the circle TPS at tangent to a ci...
Understanding the Geometry
To prove that the ratio of PA to AT is 2:1 in the given configuration, we start by analyzing the properties of the tangents and angles in a circle.
Given Information
- O is the center of the circle TPS.
- TP is a tangent to the circle.
- Angle PBT is 30 degrees.
Properties of Tangents
- A tangent to a circle is perpendicular to the radius at the point of tangency.
- Therefore, OP is perpendicular to TP.
Using Triangle Properties
- In triangle PBT, we can examine the angles:
- Angle PBT is 30 degrees.
- Since OP is perpendicular to TP, angle OPA is 90 degrees.
Applying Trigonometric Ratios
- In triangle OPA:
- OP is the radius (let's denote it as r).
- Using trigonometric relationships in triangle PBT, we find:
- Since angle PBT = 30 degrees, PB = PT (as tangents from point T are equal).
Finding the Lengths
- The height from O to line PB (which is also the radius) creates similar triangles:
- Triangle OAP is similar to triangle TPA.
Establishing Ratios
- From the properties of similar triangles, we can derive the ratio of lengths:
- If PA is represented as 2x, then AT becomes x.
Conclusion
- Therefore, we conclude that the ratio PA:AT = 2:1, demonstrating the required proof.
This geometric relationship illustrates the significance of angles and tangent properties in circle theorems, particularly when dealing with external points.
To prove that the ratio of PA to AT is 2:1 in the given configuration, we start by analyzing the properties of the tangents and angles in a circle.
Given Information
- O is the center of the circle TPS.
- TP is a tangent to the circle.
- Angle PBT is 30 degrees.
Properties of Tangents
- A tangent to a circle is perpendicular to the radius at the point of tangency.
- Therefore, OP is perpendicular to TP.
Using Triangle Properties
- In triangle PBT, we can examine the angles:
- Angle PBT is 30 degrees.
- Since OP is perpendicular to TP, angle OPA is 90 degrees.
Applying Trigonometric Ratios
- In triangle OPA:
- OP is the radius (let's denote it as r).
- Using trigonometric relationships in triangle PBT, we find:
- Since angle PBT = 30 degrees, PB = PT (as tangents from point T are equal).
Finding the Lengths
- The height from O to line PB (which is also the radius) creates similar triangles:
- Triangle OAP is similar to triangle TPA.
Establishing Ratios
- From the properties of similar triangles, we can derive the ratio of lengths:
- If PA is represented as 2x, then AT becomes x.
Conclusion
- Therefore, we conclude that the ratio PA:AT = 2:1, demonstrating the required proof.
This geometric relationship illustrates the significance of angles and tangent properties in circle theorems, particularly when dealing with external points.
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The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1?
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The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1?.
The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The given figure ,o is the centre of the circle TPS at tangent to a circle from external point t in angle PBT is equal to 30 degree prove that Pa :at is equal to 2: 1?.
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