Geometry problem - areas of triangles (contest math) The Next CEO of Stack OverflowContest Math GeometryMath contest geometry probabilitymath contest geometry proof problemMath contest geometry proof problem 2Contest Math Possible Triangles3D Geometry Contest Math Problemmath contest geometry problemInscribed and circumscribed non-regular polygonsSynthetic geometry with/without measurement vs analytic geometryRing Theoretical Method of Solving a Math Olympiad Problem

How long to clear the 'suck zone' of a turbofan after start is initiated?

What's the best way to handle refactoring a big file?

Anatomically Correct Strange Women In Ponds Distributing Swords

What does convergence in distribution "in the Gromov–Hausdorff" sense mean?

How to avoid supervisors with prejudiced views?

Are there any limitations on attacking while grappling?

Do I need to enable Dev Hub in my PROD Org?

"In the right combination" vs "with the right combination"?

How does the mv command work with external drives?

Received an invoice from my ex-employer billing me for training; how to handle?

What is "(CFMCC)" on an ILS approach chart?

Would a galaxy be visible from outside, but nearby?

How do scammers retract money, while you can’t?

Why am I allowed to create multiple unique pointers from a single object?

I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin

Is it ever safe to open a suspicious html file (e.g. email attachment)?

Solidity! Invalid implicit conversion from string memory to bytes memory requested

What flight has the highest ratio of time difference to flight time?

What was the first Unix version to run on a microcomputer?

Can we say or write : "No, it'sn't"?

Indicator light circuit

Preparing Indesign booklet with .psd graphics for print

What benefits would be gained by using human laborers instead of drones in deep sea mining?

Can I equip Skullclamp on a creature I am sacrificing?



Geometry problem - areas of triangles (contest math)



The Next CEO of Stack OverflowContest Math GeometryMath contest geometry probabilitymath contest geometry proof problemMath contest geometry proof problem 2Contest Math Possible Triangles3D Geometry Contest Math Problemmath contest geometry problemInscribed and circumscribed non-regular polygonsSynthetic geometry with/without measurement vs analytic geometryRing Theoretical Method of Solving a Math Olympiad Problem










1












$begingroup$


This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
enter image description here



I was able to solve it using coordinate geometry, both triangles have the same area. However, I do not expect 9th graders to know this method. Is there a simpler solution that I am not seeing?






contest-math euclidean-geometry







share|cite|improve this question











$endgroup$
















    1












    $begingroup$


    This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
    enter image description here



    I was able to solve it using coordinate geometry, both triangles have the same area. However, I do not expect 9th graders to know this method. Is there a simpler solution that I am not seeing?






    contest-math euclidean-geometry







    share|cite|improve this question











    $endgroup$














      1












      1








      1


      1



      $begingroup$


      This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
      enter image description here



      I was able to solve it using coordinate geometry, both triangles have the same area. However, I do not expect 9th graders to know this method. Is there a simpler solution that I am not seeing?






      contest-math euclidean-geometry







      share|cite|improve this question











      $endgroup$




      This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
      enter image description here



      I was able to solve it using coordinate geometry, both triangles have the same area. However, I do not expect 9th graders to know this method. Is there a simpler solution that I am not seeing?






      contest-math euclidean-geometry




      contest-math euclidean-geometry






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 4 hours ago







      Vasya

















      asked 4 hours ago









      VasyaVasya

      4,1351618




      4,1351618




















          1 Answer
          1






          active

          oldest

          votes


















          4












          $begingroup$

          Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac12S$.



          Since $AP=2AB$ and $AQ=3AD$, $A_triangle APQ$ is $2times 3=6$ times $A_triangle ABD$.
          Similarly $A_triangle AQR$ and $A_triangle APR$. So $A_triangle PQR = A_triangle APQ+A_triangle AQR - A_triangle APR$, giving the answer.



          All this is just the ratio of areas of triangle with same base and ratio of height (or vice versa), which a year 9 student should already know.






          share|cite|improve this answer









          $endgroup$













            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%2f3167832%2fgeometry-problem-areas-of-triangles-contest-math%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









            4












            $begingroup$

            Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac12S$.



            Since $AP=2AB$ and $AQ=3AD$, $A_triangle APQ$ is $2times 3=6$ times $A_triangle ABD$.
            Similarly $A_triangle AQR$ and $A_triangle APR$. So $A_triangle PQR = A_triangle APQ+A_triangle AQR - A_triangle APR$, giving the answer.



            All this is just the ratio of areas of triangle with same base and ratio of height (or vice versa), which a year 9 student should already know.






            share|cite|improve this answer









            $endgroup$

















              4












              $begingroup$

              Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac12S$.



              Since $AP=2AB$ and $AQ=3AD$, $A_triangle APQ$ is $2times 3=6$ times $A_triangle ABD$.
              Similarly $A_triangle AQR$ and $A_triangle APR$. So $A_triangle PQR = A_triangle APQ+A_triangle AQR - A_triangle APR$, giving the answer.



              All this is just the ratio of areas of triangle with same base and ratio of height (or vice versa), which a year 9 student should already know.






              share|cite|improve this answer









              $endgroup$















                4












                4








                4





                $begingroup$

                Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac12S$.



                Since $AP=2AB$ and $AQ=3AD$, $A_triangle APQ$ is $2times 3=6$ times $A_triangle ABD$.
                Similarly $A_triangle AQR$ and $A_triangle APR$. So $A_triangle PQR = A_triangle APQ+A_triangle AQR - A_triangle APR$, giving the answer.



                All this is just the ratio of areas of triangle with same base and ratio of height (or vice versa), which a year 9 student should already know.






                share|cite|improve this answer









                $endgroup$



                Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac12S$.



                Since $AP=2AB$ and $AQ=3AD$, $A_triangle APQ$ is $2times 3=6$ times $A_triangle ABD$.
                Similarly $A_triangle AQR$ and $A_triangle APR$. So $A_triangle PQR = A_triangle APQ+A_triangle AQR - A_triangle APR$, giving the answer.



                All this is just the ratio of areas of triangle with same base and ratio of height (or vice versa), which a year 9 student should already know.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 4 hours ago









                user10354138user10354138

                7,4722925




                7,4722925



























                    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.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3167832%2fgeometry-problem-areas-of-triangles-contest-math%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

                    The Calvary Singular or Plural The 2019 Stack Overflow Developer Survey Results Are InAre collective nouns always plural, or are certain ones singular?Is “audience” singular or plural?“Wasn't” vs. “weren't” in a vernacular sentence“My last couple of years” — singular or plural?Is 'rest' singular or plural?Is “all but one” singular or plural?Whether to use the singular or plural form of basis?Singular and Plural for numbersIs there a plural form of teeth?performance: plural vs singular?singular or plural nouns?Singular and Plural

                    Image
                    How to type a long/em dash `—` Did Scotland spend $250,000 for the slogan "Welcome to Scotland"? Worn-tile Scrabble Can we generate random numbers using irrational numbers like π and e? Geography at the pixel level For what reasons would an animal species NOT cross a *horizontal* land bridge? A female thief is not sold to make restitution -- so what happens instead? What is this sharp, curved notch on my knife for? Why does the nucleus not repel itself? What do I do when my TA workload is more than expected? How do I free up internal storage if I don't have any apps downloaded? How come people say “Would of”? Can there be female White Walkers? Cooking pasta in a water boiler If I score a critical hit on an 18 or higher, what are my chances of getting a critical hit if I roll 3d20? Is it ok to offer lower paid work as a trial period before negotiating for a full-time job? Can an undergraduate be advised by a professor who is very far...
                    Read more

                    How does one intimidate enemies without having the capacity for violence?Ideas for how aliens would approach this fight?How to convey the scale of my humanoid without science or units?How does the “space drive” conserve momentum?Planet Vanishes - How does this affect the orbiting starships?How does a community of a Universe Simulator have the same language as its creator?How would US Presidential elections be affected if voters could choose the state their vote for President was counted in?How Does One Ensures the Immortality of Their ConsciousnessHow do I retain national independence while also having a one world government?How can Ganymede have an Earth-like gravity without us having realized it?How would one make a lion mount for a fantasy world?

                    Image
                    Emailing HOD to enhance faculty application Forgetting the musical notes while performing in concert Why do I get two different answers for this counting problem? Watching something be written to a file live with tail What defenses are there against being summoned by the Gate spell? Why do bosons tend to occupy the same state? I'm flying to France today and my passport expires in less than 2 months Is the Joker left-handed? How to prevent "they're falling in love" trope Why is it a bad idea to hire a hitman to eliminate most corrupt politicians? I Accidentally Deleted a Stock Terminal Theme What reasons are there for a Capitalist to oppose a 100% inheritance tax? Twin primes whose sum is a cube Can I ask the recruiters in my resume to put the reason why I am rejected? Could gravitational lensing be used to protect a spaceship from a laser? Do I have a twin with permutated remainders? How do I write bicross product symbols in latex? If...
                    Read more

                    Output visual diagram of pictureASCII-art logic gate diagramBooks on a ShelfDetermine the Dimensions of a Rotated RectangleDraw a Houndstooth PatternDraw and label an ASCII hexagonal gridGolf me an ASCII AlphabetASCII Jigsaw PuzzleOutput a pretty boxASCII-Art Venn DiagramASCII Exact Cover with Rectangles

                    Image
                    Turning a hard to access nut? Magento 2: Make category field required in product form in backend Are hand made posters acceptable in Academia? How do researchers send unsolicited emails asking for feedback on their works? Why is indicated airspeed rather than ground speed used during the takeoff roll? Perfect 4th is dissonant? If I cast the Enlarge/Reduce spell on an arrow, what weapon could it count as? What is it called when someone votes for an option that's not their first choice? Is it okay for a cleric of life to use spells like Animate Dead and/or Contagion? Would mining huge amounts of resources on the Moon change its orbit? Creating points with attributes from coordinates in ArcPy Box half filled color Asserting that Atheism and Theism are both faith based positions Why is "la Gestapo" feminine? Recursively updating the MLE as new observations stream in Why didn't Voldemort know what Grindelwald looked like? Print last inputted...
                    Read more