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From an external point P, two tangents, PA
and PB are drawn to a circle with centre O.
At a point E on the circle, a tangent is drawn
to intersect PA and PB at C and D,
respectively. If PA = 10 cm, find the
perimeter of ∆PCD.?
and PB are drawn to a circle with centre O.
At a point E on the circle, a tangent is drawn
to intersect PA and PB at C and D,
respectively. If PA = 10 cm, find the
perimeter of ∆PCD.?
Most Upvoted Answer
From an external point P, two tangents, PAand PB are drawn to a circle...
Understanding the Geometry of the Problem
To solve for the perimeter of triangle PCD, we start by analyzing the given features of the configuration involving the point P, the tangents PA and PB, and the circle with center O.
Given Data
- Length of PA = 10 cm
- Tangents PA and PB are equal in length (PA = PB)
Properties of Tangents
- Tangents drawn from an external point to a circle are equal in length.
- Therefore, PB = 10 cm as well.
Identifying the Triangle
- Triangle PCD is formed by points P, C, and D, where C and D are the points of intersection of the tangent at point E with lines PA and PB respectively.
Finding Lengths PC and PD
1. Using the Tangent-Secant Theorem:
- Since CE and DE are both tangents to the circle from point E, we can conclude that:
- PC = PE (length from P to tangent point C)
- PD = PE (length from P to tangent point D)
2. Using Right Triangle Properties:
- Since PA and PB are tangents, triangles PCA and PBD are right triangles.
- Therefore, the lengths PC and PD can be found using the right triangle properties.
Calculating Perimeter of Triangle PCD
- The perimeter (P) of triangle PCD is given by:
P = PC + PD + CD
- Since PC = PD, we can simplify the perimeter calculation:
P = 2 * PC + CD
Final Calculation
- As the exact lengths of PC and CD depend on the specific positioning of point E on the circle, without loss of generality, assume PC and PD are equal to the length of the tangents from P to point E.
- Hence, if we assume CD = 10 cm (as a typical case), the perimeter simplifies to:
P = 2 * 10 + 10 = 30 cm
Thus, the perimeter of triangle PCD is 30 cm in this scenario, assuming CD is equal to the length of the tangents.
To solve for the perimeter of triangle PCD, we start by analyzing the given features of the configuration involving the point P, the tangents PA and PB, and the circle with center O.
Given Data
- Length of PA = 10 cm
- Tangents PA and PB are equal in length (PA = PB)
Properties of Tangents
- Tangents drawn from an external point to a circle are equal in length.
- Therefore, PB = 10 cm as well.
Identifying the Triangle
- Triangle PCD is formed by points P, C, and D, where C and D are the points of intersection of the tangent at point E with lines PA and PB respectively.
Finding Lengths PC and PD
1. Using the Tangent-Secant Theorem:
- Since CE and DE are both tangents to the circle from point E, we can conclude that:
- PC = PE (length from P to tangent point C)
- PD = PE (length from P to tangent point D)
2. Using Right Triangle Properties:
- Since PA and PB are tangents, triangles PCA and PBD are right triangles.
- Therefore, the lengths PC and PD can be found using the right triangle properties.
Calculating Perimeter of Triangle PCD
- The perimeter (P) of triangle PCD is given by:
P = PC + PD + CD
- Since PC = PD, we can simplify the perimeter calculation:
P = 2 * PC + CD
Final Calculation
- As the exact lengths of PC and CD depend on the specific positioning of point E on the circle, without loss of generality, assume PC and PD are equal to the length of the tangents from P to point E.
- Hence, if we assume CD = 10 cm (as a typical case), the perimeter simplifies to:
P = 2 * 10 + 10 = 30 cm
Thus, the perimeter of triangle PCD is 30 cm in this scenario, assuming CD is equal to the length of the tangents.
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From an external point P, two tangents, PAand PB are drawn to a circle with centre O.At a point E on the circle, a tangent is drawnto intersect PA and PB at C and D,respectively. If PA = 10 cm, find theperimeter of ∆PCD.?
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From an external point P, two tangents, PAand PB are drawn to a circle with centre O.At a point E on the circle, a tangent is drawnto intersect PA and PB at C and D,respectively. If PA = 10 cm, find theperimeter of ∆PCD.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about From an external point P, two tangents, PAand PB are drawn to a circle with centre O.At a point E on the circle, a tangent is drawnto intersect PA and PB at C and D,respectively. If PA = 10 cm, find theperimeter of ∆PCD.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for From an external point P, two tangents, PAand PB are drawn to a circle with centre O.At a point E on the circle, a tangent is drawnto intersect PA and PB at C and D,respectively. If PA = 10 cm, find theperimeter of ∆PCD.?.
From an external point P, two tangents, PAand PB are drawn to a circle with centre O.At a point E on the circle, a tangent is drawnto intersect PA and PB at C and D,respectively. If PA = 10 cm, find theperimeter of ∆PCD.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about From an external point P, two tangents, PAand PB are drawn to a circle with centre O.At a point E on the circle, a tangent is drawnto intersect PA and PB at C and D,respectively. If PA = 10 cm, find theperimeter of ∆PCD.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for From an external point P, two tangents, PAand PB are drawn to a circle with centre O.At a point E on the circle, a tangent is drawnto intersect PA and PB at C and D,respectively. If PA = 10 cm, find theperimeter of ∆PCD.?.
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