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