Menu Close

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-




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

Leave a Reply

Your email address will not be published. Required fields are marked *