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Question Number 8427 by tawakalitu last updated on 10/Oct/16

In a circle PQRST center O, PQRS is a cyclic quadrilateral and T is on the circle. QS is a diameter and angle QOR is 86° if PTQ is 28° . Find the angles of the quadrilateral PQRS.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{PQRST}\:\mathrm{center}\:\mathrm{O},\:\mathrm{PQRS}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{cyclic}\:\mathrm{quadrilateral}\:\mathrm{and}\:\mathrm{T}\:\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle}. \\ $$$$\mathrm{QS}\:\mathrm{is}\:\mathrm{a}\:\mathrm{diameter}\:\mathrm{and}\:\mathrm{angle}\:\mathrm{QOR}\:\mathrm{is}\:\mathrm{86}° \\ $$$$\mathrm{if}\:\mathrm{PTQ}\:\mathrm{is}\:\mathrm{28}°\:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{quadrilateral}\:\mathrm{PQRS}. \\ $$

Answered by sandy_suhendra last updated on 11/Oct/16

if QS is diameter so ∠QRS=∠QPS=90° ∠QOR=86° ⇒ ∠OQR=∠ORQ=((180−86)/2)=47° (OQ=OR) ∠QSR=90°−47°=43° ∠PSQ=∠PTQ=28° ∠PSR=43°+28°=71° ∠PQR=180°−71°=109°

$$\mathrm{if}\:\mathrm{QS}\:\mathrm{is}\:\mathrm{diameter}\:\mathrm{so}\:\angle\mathrm{QRS}=\angle\mathrm{QPS}=\mathrm{90}° \\ $$$$\angle\mathrm{QOR}=\mathrm{86}°\:\Rightarrow\:\angle\mathrm{OQR}=\angle\mathrm{ORQ}=\frac{\mathrm{180}−\mathrm{86}}{\mathrm{2}}=\mathrm{47}°\:\:\left(\mathrm{OQ}=\mathrm{OR}\right) \\ $$$$\angle\mathrm{QSR}=\mathrm{90}°−\mathrm{47}°=\mathrm{43}° \\ $$$$\angle\mathrm{PSQ}=\angle\mathrm{PTQ}=\mathrm{28}° \\ $$$$\angle\mathrm{PSR}=\mathrm{43}°+\mathrm{28}°=\mathrm{71}° \\ $$$$\angle\mathrm{PQR}=\mathrm{180}°−\mathrm{71}°=\mathrm{109}° \\ $$

Commented by Rasheed Soomro last updated on 11/Oct/16

Commented by tawakalitu last updated on 11/Oct/16

Wow, i really appreciate. thank you sir.

$$\mathrm{Wow},\:\mathrm{i}\:\mathrm{really}\:\mathrm{appreciate}.\:\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Commented by sandy_suhendra last updated on 11/Oct/16

thank′s Rasheed for your diagram I still have a trouble to upload my picture btw, what′s apps do you use to make your diagram?

$$\mathrm{thank}'\mathrm{s}\:\mathrm{Rasheed}\:\mathrm{for}\:\mathrm{your}\:\mathrm{diagram} \\ $$$$\mathrm{I}\:\mathrm{still}\:\mathrm{have}\:\mathrm{a}\:\mathrm{trouble}\:\mathrm{to}\:\mathrm{upload}\:\mathrm{my}\:\mathrm{picture} \\ $$$$\mathrm{btw},\:\mathrm{what}'\mathrm{s}\:\mathrm{apps}\:\mathrm{do}\:\mathrm{you}\:\mathrm{use}\:\mathrm{to}\:\mathrm{make}\:\mathrm{your}\:\mathrm{diagram}? \\ $$

Commented by Rasheed Soomro last updated on 11/Oct/16

You welcome! I use android version of geogebra. Very nice app in geometry! (Although not completely free from issues.) I first use geogebra to make a diagram and then make screenshot of the diagram. I made and sent the diagram to answer the question but meanwhile your answer came and the logic was same as of mine... Felt discomfort for your trouble.

$$\mathrm{You}\:\mathrm{welcome}! \\ $$$$\mathrm{I}\:\mathrm{use}\:\mathrm{android}\:\mathrm{version}\:\mathrm{of}\:\boldsymbol{\mathrm{geogebra}}.\:\mathrm{Very} \\ $$$$\mathrm{nice}\:\mathrm{app}\:\mathrm{in}\:\mathrm{geometry}!\:\left(\mathrm{Although}\:\mathrm{not}\right. \\ $$$$\left.\mathrm{completely}\:\mathrm{free}\:\mathrm{from}\:\mathrm{issues}.\right)\:\mathrm{I}\:\mathrm{first}\:\mathrm{use} \\ $$$$\mathrm{geogebra}\:\mathrm{to}\:\mathrm{make}\:\mathrm{a}\:\mathrm{diagram}\:\mathrm{and}\:\mathrm{then} \\ $$$$\mathrm{make}\:\mathrm{screenshot}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diagram}. \\ $$$$\mathrm{I}\:\mathrm{made}\:\mathrm{and}\:\mathrm{sent}\:\:\mathrm{the}\:\mathrm{diagram}\:\mathrm{to}\:\mathrm{answer} \\ $$$$\mathrm{the}\:\mathrm{question}\:\mathrm{but}\:\mathrm{meanwhile}\:\mathrm{your}\:\mathrm{answer} \\ $$$$\mathrm{came}\:\mathrm{and}\:\mathrm{the}\:\mathrm{logic}\:\mathrm{was}\:\mathrm{same}\:\mathrm{as}\:\mathrm{of}\:\mathrm{mine}... \\ $$$$ \\ $$$$\mathrm{Felt}\:\mathrm{discomfort}\:\mathrm{for}\:\mathrm{your}\:\mathrm{trouble}. \\ $$

Commented by Tinku Tara last updated on 11/Oct/16

We are actively working on reported probelms, image decoding and font size issues are related. We will address these 2 issues along with the new symbol within next 3−4 days. For the next update in coming weeks we are working on adding drawing functionality within the app so that it will be easy to write math notes involving diagrams in editor. Thanks for your cooperation and support.

$$\mathrm{We}\:\mathrm{are}\:\mathrm{actively}\:\mathrm{working}\:\mathrm{on}\:\mathrm{reported} \\ $$$$\mathrm{probelms},\:\mathrm{image}\:\mathrm{decoding}\:\mathrm{and}\:\mathrm{font} \\ $$$$\mathrm{size}\:\mathrm{issues}\:\mathrm{are}\:\mathrm{related}. \\ $$$$\mathrm{We}\:\mathrm{will}\:\mathrm{address}\:\mathrm{these}\:\mathrm{2}\:\mathrm{issues}\:\mathrm{along} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{new}\:\mathrm{symbol}\:\mathrm{within}\:\mathrm{next} \\ $$$$\mathrm{3}−\mathrm{4}\:\mathrm{days}. \\ $$$$\mathrm{For}\:\mathrm{the}\:\mathrm{next}\:\mathrm{update}\:\mathrm{in}\:\mathrm{coming} \\ $$$$\mathrm{weeks}\:\mathrm{we}\:\mathrm{are}\:\mathrm{working}\:\mathrm{on}\:\mathrm{adding} \\ $$$$\mathrm{drawing}\:\mathrm{functionality} \\ $$$$\mathrm{within}\:\mathrm{the}\:\mathrm{app}\:\mathrm{so}\:\mathrm{that}\:\mathrm{it}\:\mathrm{will} \\ $$$$\mathrm{be}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{write}\:\mathrm{math}\:\mathrm{notes} \\ $$$$\mathrm{involving}\:\mathrm{diagrams}\:\mathrm{in}\:\mathrm{editor}. \\ $$$$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{your}\:\mathrm{cooperation} \\ $$$$\mathrm{and}\:\mathrm{support}. \\ $$

Commented by Rasheed Soomro last updated on 11/Oct/16

Thanks for so much care!

$$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{so}\:\mathrm{much}\:\:\mathrm{care}! \\ $$

Commented by sandy_suhendra last updated on 12/Oct/16

Thank′s for your attention. I will wait for the best performance of this app.

$$\mathrm{Thank}'\mathrm{s}\:\mathrm{for}\:\mathrm{your}\:\mathrm{attention}.\:\mathrm{I}\:\mathrm{will}\:\mathrm{wait}\:\mathrm{for} \\ $$$$\mathrm{the}\:\mathrm{best}\:\mathrm{performance}\:\mathrm{of}\:\mathrm{this}\:\mathrm{app}. \\ $$$$ \\ $$

Commented by sandy_suhendra last updated on 12/Oct/16

Thank′s for your information

$$\mathrm{Thank}'\mathrm{s}\:\mathrm{for}\:\mathrm{your}\:\mathrm{information} \\ $$

Commented by sandy_suhendra last updated on 16/Oct/16

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