Q. 184.3( 29 Votes )

# If the angle betw

2r

Given - BPA = 60⁰

To find - length of OP

Properties -

1. Lengths of the two tangents drawn from an external point to a circle are equal.

2. In 30⁰-60⁰-90⁰ triangles, the side opposite to 30⁰ is half of the hypotenuse and the side opposite to 60⁰ is times the hypotenuse.

3. The radius from the centre of the circle to the point of tangency is perpendicular to the tangent line.

As we know that lengths of tangents drawn from the same external point to the circle are same.

PA = PB ………(1)

The angle between the two tangents is 60⁰.

BPA = 60⁰ ………(2)

Therefore, for ∆POB & ∆POA,

PB = PA ………from (1)

OB = OA = r ………..radii of same circle

And OP is the common side.

………by SSS test of similarity

BPO = APO ……….corresponding angles of similar triangles ……….(3)

But BPA = BPO + APO

60⁰ = BPO + BPO ……….from (2) & (3)

60⁰ = 2 . BPO

BPO = 30⁰ = APO

As tangents are always perpendicular to the radius

PBO = PAO = 90⁰

Therefore, ∆POB & ∆POA are 30⁰-60⁰-90⁰.

Therefore, by the property of 30⁰-60⁰-90⁰ triangle, the side opposite to 30⁰ is half of hypotenuse.

In ∆POB, BPO = 30⁰ and side opposite to BPO is OB, therefore,

OP = 2r

Hence, the length of OP is 2r units.

OR

6 cm

In the figure below, EF is the chord for outer circle, and it is tangent for inner circle at point D.

Given – radius of inner circle = AD = 4 cm

Radius of outer circle = AE = AF = 5 cm

To Find – length of chord EF

Properties –

1. The radius from the centre of the circle to the point of tangency is perpendicular to the tangent line.

As radius from the centre of the circle to the point of tangency is perpendicular to the tangent line.

AE = AF = 5 cm ………(2) (given)

AD = 4 cm ………(3) (given)

In right angled triangle ADE,

By Pythagoras Theorem,

AE2 = AD2 + ED2

52 = 42 + ED2

25 = 16 + ED2

ED2 = 9

ED = 3 ………(4)

Similarly, in right angled triangle ADF,

AF2 = AD2 + FD2

52 = 42 + FD2

25 = 16 + FD2

FD2 = 9

FD = 3 ………(5)

Now, EF = ED + FD

EF = 3 + 3 = 6 cm

Therefore, length of chord is 6 cm.

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