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Line Perpendicular to a Chord from the Center of the Circle

In the figure...

Question

In the figure, find $∠ C O E ?$

30^{\circ}

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60^{\circ}

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45^{\circ}

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75^{\circ}

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Solution

The correct option is B $60^{\circ}$
Given, $C E = E D$
$\Rightarrow E$ is the mid point of $C D .$
$∴ O E ⊥ C D \Rightarrow ∠ O E C = 90^{\circ}$

Applying angle sum property in $△ O C E$
$∠ C O E + ∠ O E C + ∠ O C E = 180^{\circ}$
$\Rightarrow ∠ C O E + 90^{\circ} + 30^{\circ} = 180^{\circ} \Rightarrow ∠ C O E = 180^{\circ} - 120^{\circ} = 60^{\circ}$

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In the figure, find ∠COE ?

The correct option is B 60∘Given, CE=ED ⇒E is the mid point of CD. ∴OE⊥CD⇒∠OEC=90∘ Applying angle sum property in △OCE ∠COE+∠OEC+∠OCE=180∘ ⇒∠COE+90∘+30∘=180∘⇒∠COE=180∘−120∘=60∘

In the figure, find $∠ C O E ?$