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प्रश्न
In the adjoining figure, O is the center of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then
(i) What is the length of each tangent segment?
(ii) What is the measure of ∠MRO?
(iii) What is the measure of ∠MRN?
उत्तर
seg RM and seg RN are tangents to the circle with center O. ....[Given]
∴ ∠OMR = ∠ONR = 90° ...[Tangent theorem]
(i) In ∆OMR,
∠OMR = 90°
∴ OR2 = OM2 + RM2 ...[Pythagoras theorem]
∴ 102 = 52 + RM2
∴ 100 = 25 + RM2
∴ RM2 = 75
∴ RM = `sqrt(75)` ...[Taking square root of both sides]
= `5sqrt(3)` cm
∴ RM = RN ......[Tangent segment theorem]
∴ Length of each tangent segment is `5sqrt(3)` cm.
(ii) In ∆RMO,
∠OMR = 90° ...[Tangent theorem]
OM = 5 cm and OR = 10 cm
∴ OM = `1/2` OR
∴ ∠MRO = 30° .....(i) [Converse of 30°–60°–90° theorem]
Similarly, ∠NRO = 30°
(iii) But, ∠MRN = ∠MRO + ∠NRO ...[Angle addition property]
= 30° + 30° ...[From (i)]
∴ ∠MRN = 60°.
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संबंधित प्रश्न
In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then
- What is the length of each tangent segment?
- What is the measure of ∠MRO?
- What is the measure of ∠MRN?
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Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements are true ? (i) It is not possible that ∠XYZ is an acute angle. (ii) ∠XYZ can’t be a right angle. (iii) ∠XYZ is an obtuse angle. (iv) Can’t make a definite statement for measure of ∠XYZ.
In the given figure, M is the centre of the circle and seg KL is a tangent segment.
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In the given figure, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle is r and l(AB) = r, Prove that ▢ABOC is a square. 
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∴ Quadrilateral ABOC is `square`
Similarly, ∠OBA = `square` ......[Tangent Theorem]
If one angle of `square` is right angle, then it is a square.
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arc APC ≅ arc DQE
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side BC ≅ side `square` .....`square`
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Given: `square`
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seg PA ≅ `square` .....[Radii of the same circle]
seg AD ≅ seg AD ......[`square`]
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