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प्रश्न
Prove the following theorem:
Tangent segments drawn from an external point to the circle are congruent.
उत्तर
Given: A is the center of the circle. Tangents through external point D touch the circle at the points P and Q.
To prove: seg DP ≅ seg DQ
Construction: Draw seg AP and seg AQ.
Proof:
In ΔPAD and ΔQAD,
seg PA ≅ seg QA ...[Radii of the same circle]
seg AD ≅ seg AD ...[Common side]
∠APD = ∠AQD = 90° ...[Tangent theorem]
∴ ΔPAD ≅ ΔQAD ...[By Hypotenuse side test]
∴ seg DP ≅ seg DQ ...[Corresponding sides of congruent triangles]
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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.
If MK = 12, KL = \[6\sqrt{3}\] then find –
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In the following figure ‘O’ is the centre of the circle.
∠AOB = 1100, m(arc AC) = 450.
Use the information and fill in the boxes with proper numbers.
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The perpendicular height of a cone is 12 cm and its slant height is 13 cm. Find the radius of the base of the cone.
In the given figure, M is the centre of the circle and seg KL is a tangent segment. L is a point of contact. If MK = 12, KL = `6sqrt3`, then find the radius of the circle.
The chords corresponding to congruent arcs of a circle are congruent. Prove the theorem by completing following activity.
Given: In a circle with centre B
arc APC ≅ arc DQE
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Proof: In ΔABC and ΔDBE,
side AB ≅ side DB ......`square`
side BC ≅ side `square` .....`square`
∠ABC ≅ ∠DBE ......[Measure of congruent arcs]
∆ABC ≅ ∆DBE ......`square`
Length of a tangent segment drawn from a point which is at a distance 15 cm from the centre of a circle is 12 cm, find the diameter of the circle?
Tangent segments drawn from an external point to a circle are congruent, prove this theorem. Complete the following activity.
Given: `square`
To Prove: `square`
Proof: Draw radius AP and radius AQ and complete the following proof of the theorem.
In ∆PAD and ∆QAD,
seg PA ≅ `square` .....[Radii of the same circle]
seg AD ≅ seg AD ......[`square`]
∠APD ≅ ∠AQD = 90° .....[Tangent theorem]
∴ ∆PAD ≅ ∆QAD ....[`square`]
∴ seg DP ≅ seg DQ .....[`square`]
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?
In the adjoining figure circle with Centre, Q touches the sides of ∠MPN at M and N. If ∠MPN = 40°, find measure of ∠MQN.
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Proof: In ∆RMO and ∆RNO,
∠RMO ≅ ∠RNO = 90° ......[`square`]
hypt OR ≅ hypt OR ......[`square`]
seg OM ≅ seg `square` ......[Radii of the same circle]
∴ ∆RMO ≅ ∆RNO ......[`square`]
∠MOR ≅ ∠NOR
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