The radius of a circle is 17 cm and the length of perpendicular drawn from its centre to chord is 8 cm. Calculate the length of the chord. if another chord of length is drawn in the same circle ,what would be it's distance from the centre
Answers
Step-by-step explanation:
Equal chords of a circle ( or congruent circles ) are equidistant from the centre (or centres).
\begin{gathered} \\ \large \mathcal \color{purple} \underline{Given}{:-}\end{gathered}
Given
:−
AB and CD are two equal chords of a circle.
where, AB = CD and OL ⊥ AB and OM ⊥ CD.
⠀⠀
\large \mathcal \color{purple} \underline{To \: \: prove}{:-}
Toprove
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Chord AB and CD are equidistant from the centre O, i.e. OL = OM.
⠀⠀⠀
\large \mathcal \color{purple} \underline{Construction}{:-}
Construction
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Join OA and OC at point O.
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\large \mathcal \color{purple} \underline{Proof}{:-}
Proof
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∵ The perpendicular from the centre of a chord bisects the chord. (Theorem 10.3)
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Therefore, ⠀⠀⠀⠀⠀⠀⠀
\bf{OL ⊥ AB \implies \: AL = \dfrac{1}{2}AB \: \: — \: eq.(i)}OL⊥AB⟹AL=
2
1
AB—eq.(i)
\bf{and, \: OM ⊥ CD \implies CM=\frac{1}{2}CD \: \: — eq.(ii)}and,OM⊥CD⟹CM=
2
1
CD—eq.(ii)
\begin{gathered} \\ \bf{But, \: \: \: \: \: AB = CD}\end{gathered}
But,AB=CD
\bf{\implies \dfrac{1}{2}AB = \dfrac{1}{2}CD}⟹
2
1
AB=
2
1
CD
\bf{\implies AL = CM \: \: \: \: [ using \: eq.(i) \: and \: (ii) ] \: \: — eq. (iii) }⟹AL=CM[usingeq.(i)and(ii)]—eq.(iii)
\begin{gathered} \\ \end{gathered}
Now, in right ∆s OAL and OCM, we have
⠀⠀⠀OA = OC ⠀⠀⠀( Radii of same circle )
⠀⠀⠀ AL = CM ⠀⠀ [ From eq. (iii) ]
and, ∠ALO = ∠ CMO ⠀⠀ ( Each 90° )
⠀⠀
⠀⠀⠀ So, ∆OAL ≅ ∆ OCM ⠀⠀( by R.H.S rule )
Thus, OL = OM⠀⠀( C.P.C.T. )
⠀⠀
Hence proved.