What is the distance between two parallel tangents of a circle having radius 4.5 cm ? Justify your answer.
Chapter 3 – CIrcle- Text Book Solution
Practice Set 3.1 | Q 4 | Page 55
What is the distance between two parallel tangents of a circle having radius 4.5 cm ? Justify your answer.
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.
Explanation:-
Given, PQ and RS are two parallel tangents to a circle at M and N respectively.
Construct line AB through the centre O parallel to RS.
It is also given that OM = ON = 4.5.
From the tangent theorem, we know that ∠OMP = ∠ONR = 90°.
Since AB || RS, we have ∠OMP = ∠AON = 90° (corresponding angles).
Similarly, ∠ONR = ∠AOM = 90° (corresponding angles).
Therefore, ∠AON + ∠AOM = 90° + 90° = 180°.
Hence, ∠AON and ∠AOM form a linear pair. This implies that ray OM and ray ON are opposite rays, and so points M, O, N are collinear (using the definition of collinear).
Therefore, MN = OM + ON = 4.5 + 4.5 = 9 cm (using the M-O-N collinear relationship).
Thus, the distance between the two parallel tangents PQ and RS is 9 cm.
Chapter 3 – Circle – Text Book Solution
Practice set 3.1 | Q 4 | Page 55
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