Q. 184.3( 29 Votes )

# If the angle betw

Answer :

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.

Answer -

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.

Answer –

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

ADE = ADF = 90⁰ ………(1)

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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