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What is the distance between two parallel tangents of a circle having radius 4.5 cm ?. Justify your answer.
Let the centre of the circle be O. Let the tangents AB and CD touch the circle at points M and N respectively. Draw seg OM and seg ON. Through O, draw line XY || line AB Line AB || line CD " ""……."(" Given ") Line XY || line AB " ""........"(" Construction ") :. line AB || line XY || line CD" "".......(i)" /OMA=/ONC=90^(@)" ""........"(" Tangent theorem ") Line AB|| line XY and line MO is the transversal, /AMO+/MOX=180^(@)" ""........"(" Interior angles theotem ") :.90^(@)+/MOX=180^(@) :./MOX=180^(@)-90^(@) :. /MOX=90^(@) Similarly /NOX=90^(@) :./MOX+/NOX=90^(@)+90^(@)=180^(@) :./MOX " and " /NOX are adjacent as well as supplementary. :./MOX " and " /NOX form linear pair. :. ray OM and ray ON are opposite rays. :. points M, O and N are collinear. :. MN is the distance between the parallel tangents AB and CD MN= 2xx " radius "" " "......."(" Diameter is twice the radius ") :.MN=2xx 4.5 :.MN=9 cm.
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What is the distance between two parallel tangents of a circle having radius 4.5 cm ?. Justify your answer.
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Text Solution Answer
Distance between parallel tangents is 9 cm.
Solution
Let the centre of the circle be O. Let the tangents AB and CD touch the circle at points M and N respectively.
Draw seg OM and seg ON.
Through O, draw line XY || line AB
Line AB || line CD …….( Given ) …….( Given ) Line XY || line AB
........( Construction )
........( Construction )
∴ ∴
line AB || line XY || line CD
.......(i) .......(i) ∠OMA=∠ONC= 90 ∘
........( Tangent theorem )
∠OMA=∠ONC=90∘ ........( Tangent theorem )
Line AB|| line XY and line MO is the transversal,
∠AMO+∠MOX= 180 ∘
........( Interior angles theotem )
∠AMO+∠MOX=180∘ ........( Interior angles theotem )
∴ 90 ∘ +∠MOX= 180 ∘ ∴90∘+∠MOX=180∘ ∴∠MOX= 180 ∘ − 90 ∘ ∴∠MOX=180∘-90∘ ∴∠MOX= 90 ∘ ∴∠MOX=90∘ Similarly ∠NOX= 90 ∘ ∠NOX=90∘ ∴∠MOX+∠NOX= 90 ∘ + 90 ∘ = 180 ∘
∴∠MOX+∠NOX=90∘+90∘=180∘
∴∠MOX and ∠NOX ∴∠MOX and ∠NOX
are adjacent as well as supplementary.
∴∠MOX and ∠NOX ∴∠MOX and ∠NOX form linear pair. ∴ ∴
ray OM and ray ON are opposite rays.
∴ ∴
points M, O and N are collinear.
∴MN ∴MN
is the distance between the parallel tangents AB and CD
MN=2× radius .......( Diameter is twice the radius )
MN=2× radius .......( Diameter is twice the radius )
∴MN=2×4.5 ∴MN=2×4.5 ∴MN=9 ∴MN=9 cm.
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What is the distance between two parallel tangents of a circle having radius 4.5 cm?
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?
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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 ? Justify Your Answer.
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What is the distance between two parallel tangents of a circle having radius 4.5 cm ? Justify your answer.
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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.
Concept: Tangent to a Circle
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Chapter 3: Circle - Q.4
Q 3 Q 2 Q 4
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SCERT Maharashtra Question Bank 10th Standard SSC Mathematics 2 Geometry Maharashtra State Board 2021
Chapter 3 Circle Q.4 | Q 3
Balbharati Mathematics 2 Geometry 10th Standard SSC Maharashtra State Board
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