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Question
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`
Solution
In ΔABC and ΔDBE,
side AB ≅ side DB ......[Radii of the same circle]
side BC ≅ side BE .....[Radii of the same circle]
∠ABC ≅ ∠DBE ......[Measure of congruent arcs]
∴ ∆ABC ≅ ∆DBE ......[SAS test of congruency]
∴ chord AC ≅ chord DE ......[Corresponding sides of congruent triangles]
RELATED QUESTIONS
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?
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.
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
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 –
(1) Radius of the circle.
(2) Measures of ∠K and ∠M.
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.
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.
The figure ΔABC is an isosceles triangle with a perimeter of 44 cm. The sides AB and BC are congruent and the length of the base AC is 12 cm. If a circle touches all three sides as shown in the figure, then find the length of the tangent segment drawn to the circle from point B.
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.
In a parallelogram ABCD, ∠B = 105°. Determine the measure of ∠A and ∠D.
In the following figure, XY = 10 cm and LT = 4 cm. Find the length of XT.
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)
