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

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

more_vert

verified 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.