Question 12 - End-of-Chapter Exercises - Chapter 5 Class 9 - I’m Up and Down, and Round and Round (Ganita Manja

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Question 12 When two chords intersect, each of them is divided into two line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord. Given: Let AB & CD be the two equal chords intersecting at point X. ∴ AB = CD To prove: Corresponding segments are equal, i.e., AX = DX and BX = CX Proof: We draw OM ⊥ AB & ON ⊥ CD So, AM = BM = 1/2 AB & DN = CN = 1/2 CDProof: We draw OM ⊥ AB & ON ⊥ CD Now, we know that Perpendicular to the chord bisects the chord Thus, AM = BM = 𝟏/𝟐 AB & DN = CN = 𝟏/𝟐 CD Now, as AB = CD, 𝟏/𝟐 AB = 𝟏/𝟐 CD ∴ AM = DN & MB = CNNow, AB & CD are equal chords And, equal chords are equidistant from the center ∴ OM = ON In ΔOMX and ΔONX, ∠OMX = ∠ONX OX = OX OM = ON ∴ ΔOMX ≅ ΔONX Thus, by CPCT MX = NX Now, we have AM = DN …(1) MB = CN …(2) MX = NX …(4) Therefore, AX = DX & BX = CX Hence proved Adding (1) & (4) AM + MX = DN + NX AX = DX Subtracting (2) & (4) BM – MX = CN – MX BX = CX