Question

What is the distance between two parallel tangents of a circle having radius 4.5 cm? Justify your answer.

What is the distance between two parallel tangents of a circle having radius 4.5 cm?

Answer 1

In the given figure, O is the centre of the circle. Line PT and line QR are two parallel tangents to the circle at P and Q, respectively.

∴ ∠OPT + ∠OQR = 180º    ......(Sum of adjacent interior angles on the same side of the transversal is supplementary)

⇒ POQ is a straight line segment.

∴ PQ is the diameter of the circle.

PQ = Distance between the parallel tangents PT and QR

= 2 × Radius

= 2 × 4.5

= 9 cm

Thus, the distance between two parallel tangents of the circle is 9 cm.

Answer 2

Let the lines PQ and RS be the two parallel tangents to circle at M and N respectively.

Through centre O, draw line AB || line RS.

OM = ON = 4.5     ......[Given]

Line AB || line RS   ......[Construction]

Line PQ || line RS  ......[Given]

∴ Line AB || line PQ || line RS

Now, ∠OMP = ∠ONR = 90°    ......(i) [Tangent theorem]

For line PQ || line AB,

∠OMP = ∠AON = 90°   ......(ii) [Corresponding angles and from (i)]

For line RS || line AB,

∠ONR = ∠AOM = 90° (iii)   ......Corresponding angles and from (i)]

∠AON + ∠AOM = 90° + 90°    ......[From (ii) and (iii)]

∴ ∠AON + ∠AOM = 180°

∴ ∠AON and ∠AOM form a linear pair.

∴ Ray OM and ray ON are opposite rays.

∴ Points M, O, N are collinear.    ......(iv)

∴ MN = OM + ON       ......[M−O–N, From (iv)]

∴ MN = 4.5 + 4.5

∴ MN = 9 cm

∴ Distance between two parallel tangents PQ and RS is 9 cm.

Answer 3

Radius of circle = 4.5 cm

Distance between two parallel tangents = Diameter of a circle

= 2r

= 2 × 4.5

= 9 cm