Theorem of Equal Chords Subtending Angles at the Center
In the figure...
Question
In the figure given, O is the centre of the circle. AB and CD are two chords of the circle. OM is perpendicular to CD and ON is perpendicular to AB. AB=24cm,ON=5cm,OM=12cm. Find the length of chord CD
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Solution
Given: AB=24cm;ON=5cm,OM=12cm. ∵ON⊥AB N is the mid point of AB. ∴AN=242cm=12cm. Now from △ANO,
AO2=ON2+AN2 ⇒r2=52+122(∵AO=CO=r) ⇒r2=25+144
⇒r=13
So,AO=CO=13cm
From ΔCMO,
CM2=CO2−OM2
⇒CM2=132−122
⇒CM2=169−144 ⇒CM2=25 ⇒CM=5 As OM⊥CD,M is the mid point of CD. ∴CD=2CM=2×5cm=10cm.