If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = a√2. Write ‘True’ or ‘False’ and justify your answer
Solution:
Consider a circle with centre O and tangents PT and PR and angle between them is 90° and radius of the 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 = 90°
∠TPO + ∠OPR = 90°
∠TPO + ∠TPO = 90° [From 1]
∠TPO = 45°
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 45° = a/OP
1/√2 = a/OP
OP = a√2
Therefore, it is true.
✦ Try This: From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP = diameter of the circle, show that ΔAPB is equilateral.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.2 Problem 5
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = a√2. 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 90°, then OP = a√2” is true
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