If angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = a√3. Write ‘True’ or ‘False’ and justify your answer
Solution:
Consider a circle with center O and tangents PT and PR, angle between them is 60° and radius of circle is a.
In △OTP and △ORP
TO = OR [radii of same circle]
OP = OP [common]
TP = PR [tangents through an external point to a circle are equal]
△OTP ≅ △ORP [By SSS Criterion ]
∠TPO = ∠OPR [c.p.c.t] --- [1]
Given, ∠TPR = 60°
∠TPO + ∠OPR = 60°
∠TPO + ∠TPO = 60° [From equation 1]
∠TPO = 30°
As tangent at any point on the circle is perpendicular to the radius through point of contact
OT ⏊ TP
∠OTP = 90°
△POT is a right-angled triangle
We know that,
sin θ = perpendicular/hypotenuse
sin ∠TPO = OT/OP = a/OP
sin 30° = a/OP
1/2 = a/OP
OP = 2a
Therefore, the statement is false.
✦ Try This: If the angle between two tangents drawn from an external point P to a circle of radius and centre O, is 60° then find the length of OP.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.2 Problem 6
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = a√3. Write ‘True’ or ‘False’ and justify your answer
Summary:
The statement “If angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = a√3” is false
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