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प्रश्न
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
To Prove: Chord AC ≅ chord DE
Proof: In ΔABC and ΔDBE,
side AB ≅ side DB ...`square`
side BC ≅ side `square` ...`square`
∠ABC ≅ ∠DBE ...[Measure of congruent arcs]
∆ABC ≅ ∆DBE ...`square`
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उत्तर
Given: In a circle with centre B
arc APC ≅ arc DQE
To Prove: Chord AC ≅ chord DE
Proof: In ΔABC and ΔDBE,
side AB ≅ side DB ...\[\boxed{\text{[Radii of the same circle]}}\]
side BC ≅ side \[\boxed{\text{BE}}\] ...\[\boxed{\text{[Radii of the same circle]}}\]
∠ABC ≅ ∠DBE ...[Measure of congruent arcs]
∴ ∆ABC ≅ ∆DBE ...\[\boxed{\text{[SAS test of congruency]}}\]
∴ chord AC ≅ chord DE ...[Corresponding sides of congruent triangles]
संबंधित प्रश्न
In the given figure, the circles with centres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm. 
In the given figure, O is the centre of the circle and B is a point of contact. seg OE ⊥ seg AD, AB = 12, AC = 8, find (1) AD (2) DC (3) DE.
In the given figure, seg EF is a diameter and seg DF is a tangent segment. The radius of the circle is r. Prove that, DE × GE = 4r2
Four alternative answers for the following question is given. Choose the correct alternative.
Length of a tangent segment drawn from a point which is at a distance 12.5 cm from the centre of a circle is 12 cm, find the diameter of the circle.
Four alternative answers for the following question is given. Choose the correct alternative.
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, 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. 
Proof: Draw segment OB and OC.
`l`(AB) = r ...[Given] (i)
AB = AC ...[`square`] (ii)
But OB = OC = r ...[`square`] (iii)
From (i), (ii) and (iii)
AB = `square` = OB = OC = r
∴ Quadrilateral ABOC is `square`
Similarly, ∠OBA = `square` ...[Tangent Theorem]
If one angle of `square` is right angle, then it is a square.
∴ Quadrilateral ABOC is a square.
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.
Segment DP and segment DQ are tangent segments to the circle with center A. If DP = 7 cm. So find the length of the segment DQ?
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
- What is the length of each tangent segment?
- What is the measure of ∠MRO?
- What is the measure of ∠MRN?
If AB and CD are the common tangents in the circles of two unequal (different) radii, then show that seg AB ≅ seg CD.
Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON with the help of activity.
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
Similairy ∠MRO ≅ `square` ...[`square`]
Prove that, tangent segments drawn from an external point to the circle are congruent.
A circle touches side BC at point P of the ΔABC, from outside of the triangle. Further extended lines AC and AB are tangents to the circle at N and M respectively. Prove that : AM = `1/2` (Perimeter of ΔABC)
