Geometry problem - areas of triangles (contest math) The Next CEO of Stack OverflowContest...
Novel about a guy who is possessed by the divine essence and the world ends?
What happened in Rome, when the western empire "fell"?
Indicator light circuit
Should I tutor a student who I know has cheated on their homework?
Is micro rebar a better way to reinforce concrete than rebar?
Why don't programming languages automatically manage the synchronous/asynchronous problem?
How does the mv command work with external drives?
How to invert MapIndexed on a ragged structure? How to construct a tree from rules?
What is "(CFMCC)" on an ILS approach chart?
I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin
Why didn't Khan get resurrected in the Genesis Explosion?
Received an invoice from my ex-employer billing me for training; how to handle?
How do I go from 300 unfinished/half written blog posts, to published posts?
What happens if you roll doubles 3 times then land on "Go to jail?"
In excess I'm lethal
Is "for causing autism in X" grammatical?
Which kind of appliances can one connect to electric sockets located in an airplane's toilet?
Preparing Indesign booklet with .psd graphics for print
Make solar eclipses exceedingly rare, but still have new moons
What was the first Unix version to run on a microcomputer?
What does convergence in distribution "in the Gromov–Hausdorff" sense mean?
If Nick Fury and Coulson already knew about aliens (Kree and Skrull) why did they wait until Thor's appearance to start making weapons?
Is there a difference between "Fahrstuhl" and "Aufzug"
Do I need to enable Dev Hub in my PROD Org?
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
$begingroup$
This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
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
asked 4 hours ago
VasyaVasya
4,1351618
$endgroup$
add a comment |
$begingroup$
This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
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
asked 4 hours ago
VasyaVasya
4,1351618
$endgroup$
add a comment |
$begingroup$
This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
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
asked 4 hours ago
VasyaVasya
4,1351618
$endgroup$
This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29.
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
asked 4 hours ago
VasyaVasya
4,1351618
asked 4 hours ago
VasyaVasya
4,1351618
edited 3 hours ago
Vasya
asked 4 hours ago
VasyaVasya
4,1351618
asked 4 hours ago
VasyaVasya
4,1351618
asked 4 hours ago
VasyaVasya
4,1351618
4,1351618
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac{1}{2}S$.
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.
answered 4 hours ago
user10354138user10354138
7,4722925
$endgroup$
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
$begingroup$
Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac{1}{2}S$.
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.
answered 4 hours ago
user10354138user10354138
7,4722925
$endgroup$
add a comment |
$begingroup$
Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac{1}{2}S$.
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.
answered 4 hours ago
user10354138user10354138
7,4722925
$endgroup$
add a comment |
$begingroup$
Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac{1}{2}S$.
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.
answered 4 hours ago
user10354138user10354138
7,4722925
$endgroup$
Since $D$ is the midpoint of $BC$, $A_triangle ACD=A_triangle ABD=frac{1}{2}S$.
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.
answered 4 hours ago
user10354138user10354138
7,4722925
answered 4 hours ago
user10354138user10354138
7,4722925
answered 4 hours ago
user10354138user10354138
7,4722925
answered 4 hours ago
user10354138user10354138
7,4722925
7,4722925
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
