Geometrical Applications of Complex Numbers












1















The complex numbers $z_1,z_2,z_3$ satisfying $$dfrac{z_1+z_3}{z_2-z_3}=dfrac{1-isqrt{3}}{2}$$ are the vertices of a triangle which is:



a) of area $0$



b) equilateral



c) right angled and isosceles



d) obtuse angled




$$$$



All I got was that from the Rotation Theorem, $$argleft(dfrac{z_1+z_3}{z_2-z_3}right)=-pi/3$$ and that $$|z_1+z_3|=|z_2-z_3|$$
$$$$
Could somebody please show me how to solve this problem? Many thanks!






calculus complex-analysis algebra-precalculus geometry complex-numbers







share|cite|improve this question




















  • 1




    If the numerator were $z_1-z_3$ instead of $z_1+z_3$, then the answer would be (b), because the information you're given would imply that two of the sides have equal length and the angle between those two sides is $60^circ$.
    – grand_chat
    Mar 25 '16 at 7:04










  • Hint: Shift the origin to any of $z_1, z_2, z_3$ to see the properties of triangle, as angles and sides are invariant under translation of origin.
    – mea43
    Mar 25 '16 at 7:10












  • You should edit your post: in fact it's $frac{text{z1}+text{z3}}{text{z2}-text{z3}}=e^{-ifrac{pi }{3}}$ and $arg left(frac{text{z1}+text{z3}}{text{z2}-text{z3}}right)=-frac{pi }{3}$. As @user21820, I don't think you can know much more about $z1,z2,z3$...
    – Eddy Khemiri
    Mar 25 '16 at 10:36










  • @mea43 Sir I tried for very long, but couldn't reach anywhere. Could you please help me?
    – Better World
    Mar 25 '16 at 11:01












  • @EddyKhemiri Edited, thanks for informing me.
    – Better World
    Mar 25 '16 at 11:03


















1















The complex numbers $z_1,z_2,z_3$ satisfying $$dfrac{z_1+z_3}{z_2-z_3}=dfrac{1-isqrt{3}}{2}$$ are the vertices of a triangle which is:



a) of area $0$



b) equilateral



c) right angled and isosceles



d) obtuse angled




$$$$



All I got was that from the Rotation Theorem, $$argleft(dfrac{z_1+z_3}{z_2-z_3}right)=-pi/3$$ and that $$|z_1+z_3|=|z_2-z_3|$$
$$$$
Could somebody please show me how to solve this problem? Many thanks!






calculus complex-analysis algebra-precalculus geometry complex-numbers







share|cite|improve this question




















  • 1




    If the numerator were $z_1-z_3$ instead of $z_1+z_3$, then the answer would be (b), because the information you're given would imply that two of the sides have equal length and the angle between those two sides is $60^circ$.
    – grand_chat
    Mar 25 '16 at 7:04










  • Hint: Shift the origin to any of $z_1, z_2, z_3$ to see the properties of triangle, as angles and sides are invariant under translation of origin.
    – mea43
    Mar 25 '16 at 7:10












  • You should edit your post: in fact it's $frac{text{z1}+text{z3}}{text{z2}-text{z3}}=e^{-ifrac{pi }{3}}$ and $arg left(frac{text{z1}+text{z3}}{text{z2}-text{z3}}right)=-frac{pi }{3}$. As @user21820, I don't think you can know much more about $z1,z2,z3$...
    – Eddy Khemiri
    Mar 25 '16 at 10:36










  • @mea43 Sir I tried for very long, but couldn't reach anywhere. Could you please help me?
    – Better World
    Mar 25 '16 at 11:01












  • @EddyKhemiri Edited, thanks for informing me.
    – Better World
    Mar 25 '16 at 11:03
















1












1








1


2






The complex numbers $z_1,z_2,z_3$ satisfying $$dfrac{z_1+z_3}{z_2-z_3}=dfrac{1-isqrt{3}}{2}$$ are the vertices of a triangle which is:



a) of area $0$



b) equilateral



c) right angled and isosceles



d) obtuse angled




$$$$



All I got was that from the Rotation Theorem, $$argleft(dfrac{z_1+z_3}{z_2-z_3}right)=-pi/3$$ and that $$|z_1+z_3|=|z_2-z_3|$$
$$$$
Could somebody please show me how to solve this problem? Many thanks!






calculus complex-analysis algebra-precalculus geometry complex-numbers







share|cite|improve this question
















The complex numbers $z_1,z_2,z_3$ satisfying $$dfrac{z_1+z_3}{z_2-z_3}=dfrac{1-isqrt{3}}{2}$$ are the vertices of a triangle which is:



a) of area $0$



b) equilateral



c) right angled and isosceles



d) obtuse angled




$$$$



All I got was that from the Rotation Theorem, $$argleft(dfrac{z_1+z_3}{z_2-z_3}right)=-pi/3$$ and that $$|z_1+z_3|=|z_2-z_3|$$
$$$$
Could somebody please show me how to solve this problem? Many thanks!






calculus complex-analysis algebra-precalculus geometry complex-numbers




calculus complex-analysis algebra-precalculus geometry complex-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 6 at 11:13









user376343

2,7782822




2,7782822










asked Mar 25 '16 at 6:17









Better World

10018




10018








  • 1




    If the numerator were $z_1-z_3$ instead of $z_1+z_3$, then the answer would be (b), because the information you're given would imply that two of the sides have equal length and the angle between those two sides is $60^circ$.
    – grand_chat
    Mar 25 '16 at 7:04










  • Hint: Shift the origin to any of $z_1, z_2, z_3$ to see the properties of triangle, as angles and sides are invariant under translation of origin.
    – mea43
    Mar 25 '16 at 7:10












  • You should edit your post: in fact it's $frac{text{z1}+text{z3}}{text{z2}-text{z3}}=e^{-ifrac{pi }{3}}$ and $arg left(frac{text{z1}+text{z3}}{text{z2}-text{z3}}right)=-frac{pi }{3}$. As @user21820, I don't think you can know much more about $z1,z2,z3$...
    – Eddy Khemiri
    Mar 25 '16 at 10:36










  • @mea43 Sir I tried for very long, but couldn't reach anywhere. Could you please help me?
    – Better World
    Mar 25 '16 at 11:01












  • @EddyKhemiri Edited, thanks for informing me.
    – Better World
    Mar 25 '16 at 11:03
















  • 1




    If the numerator were $z_1-z_3$ instead of $z_1+z_3$, then the answer would be (b), because the information you're given would imply that two of the sides have equal length and the angle between those two sides is $60^circ$.
    – grand_chat
    Mar 25 '16 at 7:04










  • Hint: Shift the origin to any of $z_1, z_2, z_3$ to see the properties of triangle, as angles and sides are invariant under translation of origin.
    – mea43
    Mar 25 '16 at 7:10












  • You should edit your post: in fact it's $frac{text{z1}+text{z3}}{text{z2}-text{z3}}=e^{-ifrac{pi }{3}}$ and $arg left(frac{text{z1}+text{z3}}{text{z2}-text{z3}}right)=-frac{pi }{3}$. As @user21820, I don't think you can know much more about $z1,z2,z3$...
    – Eddy Khemiri
    Mar 25 '16 at 10:36










  • @mea43 Sir I tried for very long, but couldn't reach anywhere. Could you please help me?
    – Better World
    Mar 25 '16 at 11:01












  • @EddyKhemiri Edited, thanks for informing me.
    – Better World
    Mar 25 '16 at 11:03










1




1




If the numerator were $z_1-z_3$ instead of $z_1+z_3$, then the answer would be (b), because the information you're given would imply that two of the sides have equal length and the angle between those two sides is $60^circ$.
– grand_chat
Mar 25 '16 at 7:04




If the numerator were $z_1-z_3$ instead of $z_1+z_3$, then the answer would be (b), because the information you're given would imply that two of the sides have equal length and the angle between those two sides is $60^circ$.
– grand_chat
Mar 25 '16 at 7:04












Hint: Shift the origin to any of $z_1, z_2, z_3$ to see the properties of triangle, as angles and sides are invariant under translation of origin.
– mea43
Mar 25 '16 at 7:10






Hint: Shift the origin to any of $z_1, z_2, z_3$ to see the properties of triangle, as angles and sides are invariant under translation of origin.
– mea43
Mar 25 '16 at 7:10














You should edit your post: in fact it's $frac{text{z1}+text{z3}}{text{z2}-text{z3}}=e^{-ifrac{pi }{3}}$ and $arg left(frac{text{z1}+text{z3}}{text{z2}-text{z3}}right)=-frac{pi }{3}$. As @user21820, I don't think you can know much more about $z1,z2,z3$...
– Eddy Khemiri
Mar 25 '16 at 10:36




You should edit your post: in fact it's $frac{text{z1}+text{z3}}{text{z2}-text{z3}}=e^{-ifrac{pi }{3}}$ and $arg left(frac{text{z1}+text{z3}}{text{z2}-text{z3}}right)=-frac{pi }{3}$. As @user21820, I don't think you can know much more about $z1,z2,z3$...
– Eddy Khemiri
Mar 25 '16 at 10:36












@mea43 Sir I tried for very long, but couldn't reach anywhere. Could you please help me?
– Better World
Mar 25 '16 at 11:01






@mea43 Sir I tried for very long, but couldn't reach anywhere. Could you please help me?
– Better World
Mar 25 '16 at 11:01














@EddyKhemiri Edited, thanks for informing me.
– Better World
Mar 25 '16 at 11:03






@EddyKhemiri Edited, thanks for informing me.
– Better World
Mar 25 '16 at 11:03












1 Answer
1






active

oldest

votes


















1














There is a problem with the problem. By what you have stated, we know that $(-z_1,z_2,z_3)$ are the vertices of an isosceles triangle with equal sides meeting at $120^circ$. But then we don't know anything much about $(z_1,z_2,z_3)$, because as you can see translating $(-z_1,z_2,z_3)$ around makes $(z_1,z_2,z_3)$ change shape.






share|cite|improve this answer





















    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1712635%2fgeometrical-applications-of-complex-numbers%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1














    There is a problem with the problem. By what you have stated, we know that $(-z_1,z_2,z_3)$ are the vertices of an isosceles triangle with equal sides meeting at $120^circ$. But then we don't know anything much about $(z_1,z_2,z_3)$, because as you can see translating $(-z_1,z_2,z_3)$ around makes $(z_1,z_2,z_3)$ change shape.






    share|cite|improve this answer


























      1














      There is a problem with the problem. By what you have stated, we know that $(-z_1,z_2,z_3)$ are the vertices of an isosceles triangle with equal sides meeting at $120^circ$. But then we don't know anything much about $(z_1,z_2,z_3)$, because as you can see translating $(-z_1,z_2,z_3)$ around makes $(z_1,z_2,z_3)$ change shape.






      share|cite|improve this answer
























        1












        1








        1






        There is a problem with the problem. By what you have stated, we know that $(-z_1,z_2,z_3)$ are the vertices of an isosceles triangle with equal sides meeting at $120^circ$. But then we don't know anything much about $(z_1,z_2,z_3)$, because as you can see translating $(-z_1,z_2,z_3)$ around makes $(z_1,z_2,z_3)$ change shape.






        share|cite|improve this answer












        There is a problem with the problem. By what you have stated, we know that $(-z_1,z_2,z_3)$ are the vertices of an isosceles triangle with equal sides meeting at $120^circ$. But then we don't know anything much about $(z_1,z_2,z_3)$, because as you can see translating $(-z_1,z_2,z_3)$ around makes $(z_1,z_2,z_3)$ change shape.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 25 '16 at 9:09









        user21820

        38.6k542153




        38.6k542153






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1712635%2fgeometrical-applications-of-complex-numbers%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Bressuire

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more

            Cabo Verde

            Image
              Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
            Read more

            Gyllenstierna

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more