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Geometry Level 3

T P TP is a line that tangent to a circle centered at O O . If P Q T O PQ \parallel TO and O T P = 2 5 \angle {OTP} = 25^\circ , find the measure of P O Q \angle{POQ} in degrees.

Note: The diagram above is not drawn to scale.

5 0 50^\circ 2 5 25^\circ 5 7 57^\circ 6 5 65^\circ

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2 solutions

Michael Fuller
May 5, 2016

Q R P = Q P S P O Q = 50 ° \angle QRP = \angle QPS \Rightarrow \angle POQ = \large \color{#20A900}{\boxed{50°}}

It's beautiful

Jason Chrysoprase - 5 years, 1 month ago

The diagram above is drawn to scale \large \text{The diagram above is drawn to scale}

O P T is obviously 9 0 because it’s tangent to a circle \Large \angle {OPT} \space \text{is obviously}\space 90^\circ \text{ because it's tangent to a circle}

T O P = 18 0 O T P O P T = 18 0 2 5 9 0 = 6 5 \Large \angle { TOP} =\Large 180^\circ - \measuredangle{OTP} - \measuredangle{OPT} \\ \Large = 180^\circ - 25^\circ - 90^\circ \\ \Large = 65^\circ

Since P Q is parallel to C D , so we can conclude that T O P = O P Q 6 5 = O P Q \Large \text{ Since }PQ\text{ is parallel to} \space CD \text{, so we can conclude that} \\ \Large \measuredangle {TOP} = \angle {OPQ} \\ \Large 65^\circ = \measuredangle {OPQ}

Since O P and O Q is the same length, we can conclude that O P Q = O Q P 6 5 = O Q P \Large \text{Since } OP\text{ and} \space OQ \text{ is the same length, we can conclude that }\\ \Large \measuredangle {OPQ} = \angle{ OQP} \\ \Large 65^\circ = \measuredangle {OQP}

Finally, P O Q = 18 0 O P Q O Q P = 18 0 6 5 6 5 = 18 0 13 0 = 5 0 \Large \text{Finally,}\\ \Large \angle {POQ} = 180^\circ - \measuredangle{OPQ} - \measuredangle{OQP} \\ \Large = 180^\circ - 65^\circ - 65^\circ \\ \Large =180^\circ - 130^\circ \\ \Large = \Huge \color{#20A900}{\clubsuit} \color{#D61F06}{\spadesuit} \space \color{#D61F06}{50^\circ} \color{#20A900}{\clubsuit} \color{#D61F06}{\spadesuit}

The Spadesuit( )and the Clubsuit( ) are just for the decoration. Want more ? :) \large \text{The Spadesuit(} \spadesuit \text{)and the Clubsuit(} \clubsuit \text{) are just for the decoration.}\\ \large \text{Want more ? :) }

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