Practice Set 3.3 Geometry 10th Std Maths Part 2 Answers Chapter 3 Circle
Maharashtra Board Class 10 Maths Chapter 3 Circle Intext Questions and Activities
Question 2
Maharashtra Board Solution
Theorem: Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent (Textbook pg. no. 61) Given: O is the centre of circle, chord PQ = chord RS To prove: arc PMQ = arc RNS Proof: In ∆POQ and ∆ROS, [side PO ≅ side RO side OQ ≅ side OS] [Radii of the same circle] chord PQ ≅ chord RS [Given] ∴ ∆POQ ≅ ∆ROS [SSS test of congruency] ∴ ∠POQ ≅ ∠ROS (i) [c.a.c.t.] m(arc PMQ) = ∠POQ (ii) m(arc RNS) = ∠ROS (iii) [Definition of measure of minor arc] ∴ arc PMQ ≅ arc RNS [From (i), (ii) and (iii)] |
Question 3. Prove the two theorems on textbook pg.no.61 for congruent circles. (Textbook pg. no. 62) Theorem : The chords corresponding to congruent arcs of congruent circles are congruent Given: In congruent circles with centres B and R, arc APC ≅ arc DQE To prove: chord AC ≅ chord DE Proof: [m(arc APC) = ∠ABC (i) m(arc DQE) = ∠DRE] (ii) [Definition of measure of minor arc] arc APC ≅ arc DQE (iii) [Given] ∴ ∠ABC = ∠DRE (iv) [From (i), (ii) and (iii)] In ∆ABC and ∆DRE, [side AB ≅ side DR [Radii of congruent circles] side CB ≅ side ER] [From (iv)] ∠ABC ≅ ∠DRE ∴ ∆ABC ≅ ∆DRE [SAS test of congruency] |
Question 4. While proving the first theorem of the two, we assume that the minor arc APC and minor arc DQE are congruent. Can you prove the same theorem by assuming that corresponding major arcs congruent? (Textbook pg. no. 62) Statement: The chords corresponding to congruent major arcs of a circle are congruent. Given: B is the centre of circle. arc AXC ≅ arc DXE To prove: chord AC ≅ chord DE Proof: m(major arc) = 360° – m(minor arc) ∴ m(arc AXC) = 360° – m(arc APC) (i) m(arc DXE) = 360° – m(arc DQE) (ii) m(arc AXC) = m(arc DXE) (iii) [Given] ∴ 360° – m(arc APC) = 360°- m(arc DQE) [From (i), (ii) and (iii)] ∴ m(arc APC) = m(arc DQE) (iv) ∴ m(arc APC) = ∠ABC (v) [Definition of measure of minor arc] m(arc DQE) = ∠DBE (vi) ∴ ∠ABC = ∠DBE (vii) [From (iv), (v) and (vi)] In ∆ABC and ∆DBE, [side AB ≅ side DB Side CB ≅ side EB] [Radii of the same circle] ∠ABC ≅ ∠DBE [From (vii)] ∴ ∆ABC ≅ ∆DBE [SAS test of congruency] ∴ chord AC ≅ chord DE [c.s.c.t.] |
Question 5. i. In the second theorem, are the major arcs corresponding to congruent chords congruent? ii. Is the theorem true, when the chord PQ and chord RS are diameters of the circle? (Textbook pg. no. 62)
Solution & Step-by-Step Answer:
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Solution & Step-by-Step Answer:
i. Yes, the major arcs corresponding to congruent chords are congruent. Proof: In ∆POQ and ∆ROS, seg OP ≅ seg OR [Radii of the same circle] seg OQ ≅ seg OS [Radii of the same circle] seg PQ ≅ seg RS [Given] ∴ ∆POQ ≅ ∆ROS [SSS test of congruence] ∴ ∠POQ ≅ ∠SOR (i) [c.a.c.t] [ m(arc PMQ) = ∠POQ (ii) m(arc RNS) = ∠SOR ] (iii) [Definition of measure of minor arc] ∴ m(arc PMQ) = m(arc RNS) m(minor arc) = 360° – m(major arc) (iv) [From (i), (ii) and (iii)] m(arc PMQ) = 360° – m(arc PXQ) (v) and m(arc RNS) = 360° – m(arc RXS) (vi) ∴ 360°- m(arc PXQ) = 360°- m(arc RXS) [From (iv), (v) and (vi)] ∴ m(arc PXQ) = m(arc RXS)
ii. Yes, the major arcs corresponding to congruent chords (diameters) are congruent. |