Equation of line

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












Suppose that you don't already know the equation of a line. You take two points A and B. You join those two points and extend that line indefinitely. Now consider any point on that line and mark it as (0,0). Draw your coordinate system accordingly and consider two quantites: rA and rB.



$A_y = r_AA_x$

$B_y = r_BB_x$

rA is always equal to rB!



Consider any other point on the line. The way Y is related to X is the same again!



How is this possible? Is there any proof? Can we surely say that all points that lie on the line have their Ys related to Xs in the same way? Is there no such point which does not have the same relation of its X and Y as the other points on the same line?






geometry







share|cite|improve this question





















  • Use similar triangles. (BTW: You've rediscovered that the equation for any (non-vertical) line through the origin is $y=mx$ for some $m$. Congratulations! :) )
    – Blue
    Aug 31 at 2:56











  • Wow, didn't notice the triangles formed. Searching for such stuff didn't yield any good results. Got my answer in a matter of minutes. Thanks!
    – jp_newbie
    Aug 31 at 4:00














up vote
0
down vote

favorite












Suppose that you don't already know the equation of a line. You take two points A and B. You join those two points and extend that line indefinitely. Now consider any point on that line and mark it as (0,0). Draw your coordinate system accordingly and consider two quantites: rA and rB.



$A_y = r_AA_x$

$B_y = r_BB_x$

rA is always equal to rB!



Consider any other point on the line. The way Y is related to X is the same again!



How is this possible? Is there any proof? Can we surely say that all points that lie on the line have their Ys related to Xs in the same way? Is there no such point which does not have the same relation of its X and Y as the other points on the same line?






geometry







share|cite|improve this question





















  • Use similar triangles. (BTW: You've rediscovered that the equation for any (non-vertical) line through the origin is $y=mx$ for some $m$. Congratulations! :) )
    – Blue
    Aug 31 at 2:56











  • Wow, didn't notice the triangles formed. Searching for such stuff didn't yield any good results. Got my answer in a matter of minutes. Thanks!
    – jp_newbie
    Aug 31 at 4:00












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Suppose that you don't already know the equation of a line. You take two points A and B. You join those two points and extend that line indefinitely. Now consider any point on that line and mark it as (0,0). Draw your coordinate system accordingly and consider two quantites: rA and rB.



$A_y = r_AA_x$

$B_y = r_BB_x$

rA is always equal to rB!



Consider any other point on the line. The way Y is related to X is the same again!



How is this possible? Is there any proof? Can we surely say that all points that lie on the line have their Ys related to Xs in the same way? Is there no such point which does not have the same relation of its X and Y as the other points on the same line?






geometry







share|cite|improve this question













Suppose that you don't already know the equation of a line. You take two points A and B. You join those two points and extend that line indefinitely. Now consider any point on that line and mark it as (0,0). Draw your coordinate system accordingly and consider two quantites: rA and rB.



$A_y = r_AA_x$

$B_y = r_BB_x$

rA is always equal to rB!



Consider any other point on the line. The way Y is related to X is the same again!



How is this possible? Is there any proof? Can we surely say that all points that lie on the line have their Ys related to Xs in the same way? Is there no such point which does not have the same relation of its X and Y as the other points on the same line?






geometry




geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 31 at 2:46









jp_newbie

185




185











  • Use similar triangles. (BTW: You've rediscovered that the equation for any (non-vertical) line through the origin is $y=mx$ for some $m$. Congratulations! :) )
    – Blue
    Aug 31 at 2:56











  • Wow, didn't notice the triangles formed. Searching for such stuff didn't yield any good results. Got my answer in a matter of minutes. Thanks!
    – jp_newbie
    Aug 31 at 4:00
















  • Use similar triangles. (BTW: You've rediscovered that the equation for any (non-vertical) line through the origin is $y=mx$ for some $m$. Congratulations! :) )
    – Blue
    Aug 31 at 2:56











  • Wow, didn't notice the triangles formed. Searching for such stuff didn't yield any good results. Got my answer in a matter of minutes. Thanks!
    – jp_newbie
    Aug 31 at 4:00















Use similar triangles. (BTW: You've rediscovered that the equation for any (non-vertical) line through the origin is $y=mx$ for some $m$. Congratulations! :) )
– Blue
Aug 31 at 2:56





Use similar triangles. (BTW: You've rediscovered that the equation for any (non-vertical) line through the origin is $y=mx$ for some $m$. Congratulations! :) )
– Blue
Aug 31 at 2:56













Wow, didn't notice the triangles formed. Searching for such stuff didn't yield any good results. Got my answer in a matter of minutes. Thanks!
– jp_newbie
Aug 31 at 4:00




Wow, didn't notice the triangles formed. Searching for such stuff didn't yield any good results. Got my answer in a matter of minutes. Thanks!
– jp_newbie
Aug 31 at 4:00










1 Answer
1






active

oldest

votes

















up vote
1
down vote



accepted










enter image description here



Note that $triangle OCA$ is similar to $triangle ODB$



It follows that $dfracA_yA_x = dfracB_yB_x$






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',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    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%2f2900252%2fequation-of-line%23new-answer', 'question_page');

    );

    Post as a guest

















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    enter image description here



    Note that $triangle OCA$ is similar to $triangle ODB$



    It follows that $dfracA_yA_x = dfracB_yB_x$






    share|cite|improve this answer
























      up vote
      1
      down vote



      accepted










      enter image description here



      Note that $triangle OCA$ is similar to $triangle ODB$



      It follows that $dfracA_yA_x = dfracB_yB_x$






      share|cite|improve this answer






















        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        enter image description here



        Note that $triangle OCA$ is similar to $triangle ODB$



        It follows that $dfracA_yA_x = dfracB_yB_x$






        share|cite|improve this answer












        enter image description here



        Note that $triangle OCA$ is similar to $triangle ODB$



        It follows that $dfracA_yA_x = dfracB_yB_x$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 31 at 3:03









        steven gregory

        16.7k22155




        16.7k22155



























             

            draft saved


            draft discarded















































             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2900252%2fequation-of-line%23new-answer', 'question_page');

            );

            Post as a guest
































































            -

            這個網誌中的熱門文章

            How to combine Bézier curves to a surface?

            圖片
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
            閱讀完整內容

            Mutual Information Always Non-negative

            圖片
            Clash Royale CLAN TAG #URR8PPP up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 '12 at 15:38 Cameron Buie 83.6k 7 71 154 asked Jun 17 '12 at 15:30 Omri 359 3 13 3 Convexity of the function $tmapsto tlog t$. – Did Jun 17 '12 at 15:33 In addition, the convexity properties require the coefficients in the linear combination sum 1. Then, as p(x,y) is a probability distribution, it fullfits such condition. – Francisco Javier Delgado Ceped Aug 10 at 18:44 add a comment  |  up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 ...
            閱讀完整內容

            Why am i infinitely getting the same tweet with the Twitter Search API?

            圖片
            up vote 0 down vote favorite My code workes perfectly for other queries, but for one query I am infinitely getting the same tweet. I can't understand what the problem is. API call: query='Allergic asthma OR Nonallergic asthma OR Occupational asthma OR EIB OR Exercise-induced bronchoconstriction OR Nocturnal asthma OR Cough-variant asthma' new_tweets = api.search(q=searchQuery, count=100, since_id=since_id, lang='en',tweet_mode='extended') if not new_tweets: print("No more tweets found") break for tweet in new_tweets: if True: print(jsonpickle.encode(tweet._json, unpicklable=False)) my_file = open('searchTweets.txt','a') my_file.write(jsonpickle.encode(tweet._json, unpicklable=False)+'n') my_file.close() else: continue Output in file: https://pastebin.com/JmRCsTeE These are the JSONs of the same Tweet. Other queries that work: Breast cancer OR Prostate cancer OR Colon cancer OR Lung cancer Type 2 dia...
            閱讀完整內容