If $|MA|^2=|MB|^2+|MC|^2$. Find the $angle BMC=?$ [closed]
$begingroup$
My problem is:
The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$
Is this question related to similar triangles? Or which theorem should I apply?
geometry contest-math euclidean-geometry book-recommendation
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closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
My problem is:
The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$
Is this question related to similar triangles? Or which theorem should I apply?
geometry contest-math euclidean-geometry book-recommendation
$endgroup$
closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
My problem is:
The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$
Is this question related to similar triangles? Or which theorem should I apply?
geometry contest-math euclidean-geometry book-recommendation
$endgroup$
My problem is:
The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$
Is this question related to similar triangles? Or which theorem should I apply?
geometry contest-math euclidean-geometry book-recommendation
geometry contest-math euclidean-geometry book-recommendation
edited Dec 26 '18 at 11:56
greedoid
44k1155109
44k1155109
asked Dec 26 '18 at 11:39
ElementaryElementary
361111
361111
closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Hint:
Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?

Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...
$endgroup$
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I'm practicing the hint you're saying.
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– Elementary
Dec 26 '18 at 11:56
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Is it possible, to add picture, if you have a few time ..I could not do..:(
$endgroup$
– Elementary
Dec 26 '18 at 12:47
$begingroup$
Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
$endgroup$
– Elementary
Dec 26 '18 at 13:00
$begingroup$
I made some edit...
$endgroup$
– greedoid
Dec 26 '18 at 13:03
$begingroup$
Thank you again..
$endgroup$
– Elementary
Dec 26 '18 at 13:26
add a comment |
$begingroup$
Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
$$|MA|^2=|MB|^2+|MC|^2$$
$$implies y^2+2sqrt3ay-a^2=0$$
$$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative
Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)
$endgroup$
$begingroup$
How do you get, $sqrt3 a$
$endgroup$
– Elementary
Dec 26 '18 at 12:05
$begingroup$
Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
$endgroup$
– Ankit Kumar
Dec 26 '18 at 12:06
add a comment |
$begingroup$
That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.
$endgroup$
1
$begingroup$
No, this is wrong approach.
$endgroup$
– Elementary
Dec 26 '18 at 11:51
1
$begingroup$
The answer is incorrect too
$endgroup$
– Ankit Kumar
Dec 26 '18 at 11:52
$begingroup$
But why doesn't that work?
$endgroup$
– Viktor Glombik
Dec 26 '18 at 12:39
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?

Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...
$endgroup$
$begingroup$
I'm practicing the hint you're saying.
$endgroup$
– Elementary
Dec 26 '18 at 11:56
$begingroup$
Is it possible, to add picture, if you have a few time ..I could not do..:(
$endgroup$
– Elementary
Dec 26 '18 at 12:47
$begingroup$
Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
$endgroup$
– Elementary
Dec 26 '18 at 13:00
$begingroup$
I made some edit...
$endgroup$
– greedoid
Dec 26 '18 at 13:03
$begingroup$
Thank you again..
$endgroup$
– Elementary
Dec 26 '18 at 13:26
add a comment |
$begingroup$
Hint:
Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?

Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...
$endgroup$
$begingroup$
I'm practicing the hint you're saying.
$endgroup$
– Elementary
Dec 26 '18 at 11:56
$begingroup$
Is it possible, to add picture, if you have a few time ..I could not do..:(
$endgroup$
– Elementary
Dec 26 '18 at 12:47
$begingroup$
Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
$endgroup$
– Elementary
Dec 26 '18 at 13:00
$begingroup$
I made some edit...
$endgroup$
– greedoid
Dec 26 '18 at 13:03
$begingroup$
Thank you again..
$endgroup$
– Elementary
Dec 26 '18 at 13:26
add a comment |
$begingroup$
Hint:
Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?

Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...
$endgroup$
Hint:
Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?

Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...
edited Dec 26 '18 at 13:03
answered Dec 26 '18 at 11:53
greedoidgreedoid
44k1155109
44k1155109
$begingroup$
I'm practicing the hint you're saying.
$endgroup$
– Elementary
Dec 26 '18 at 11:56
$begingroup$
Is it possible, to add picture, if you have a few time ..I could not do..:(
$endgroup$
– Elementary
Dec 26 '18 at 12:47
$begingroup$
Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
$endgroup$
– Elementary
Dec 26 '18 at 13:00
$begingroup$
I made some edit...
$endgroup$
– greedoid
Dec 26 '18 at 13:03
$begingroup$
Thank you again..
$endgroup$
– Elementary
Dec 26 '18 at 13:26
add a comment |
$begingroup$
I'm practicing the hint you're saying.
$endgroup$
– Elementary
Dec 26 '18 at 11:56
$begingroup$
Is it possible, to add picture, if you have a few time ..I could not do..:(
$endgroup$
– Elementary
Dec 26 '18 at 12:47
$begingroup$
Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
$endgroup$
– Elementary
Dec 26 '18 at 13:00
$begingroup$
I made some edit...
$endgroup$
– greedoid
Dec 26 '18 at 13:03
$begingroup$
Thank you again..
$endgroup$
– Elementary
Dec 26 '18 at 13:26
$begingroup$
I'm practicing the hint you're saying.
$endgroup$
– Elementary
Dec 26 '18 at 11:56
$begingroup$
I'm practicing the hint you're saying.
$endgroup$
– Elementary
Dec 26 '18 at 11:56
$begingroup$
Is it possible, to add picture, if you have a few time ..I could not do..:(
$endgroup$
– Elementary
Dec 26 '18 at 12:47
$begingroup$
Is it possible, to add picture, if you have a few time ..I could not do..:(
$endgroup$
– Elementary
Dec 26 '18 at 12:47
$begingroup$
Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
$endgroup$
– Elementary
Dec 26 '18 at 13:00
$begingroup$
Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
$endgroup$
– Elementary
Dec 26 '18 at 13:00
$begingroup$
I made some edit...
$endgroup$
– greedoid
Dec 26 '18 at 13:03
$begingroup$
I made some edit...
$endgroup$
– greedoid
Dec 26 '18 at 13:03
$begingroup$
Thank you again..
$endgroup$
– Elementary
Dec 26 '18 at 13:26
$begingroup$
Thank you again..
$endgroup$
– Elementary
Dec 26 '18 at 13:26
add a comment |
$begingroup$
Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
$$|MA|^2=|MB|^2+|MC|^2$$
$$implies y^2+2sqrt3ay-a^2=0$$
$$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative
Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)
$endgroup$
$begingroup$
How do you get, $sqrt3 a$
$endgroup$
– Elementary
Dec 26 '18 at 12:05
$begingroup$
Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
$endgroup$
– Ankit Kumar
Dec 26 '18 at 12:06
add a comment |
$begingroup$
Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
$$|MA|^2=|MB|^2+|MC|^2$$
$$implies y^2+2sqrt3ay-a^2=0$$
$$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative
Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)
$endgroup$
$begingroup$
How do you get, $sqrt3 a$
$endgroup$
– Elementary
Dec 26 '18 at 12:05
$begingroup$
Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
$endgroup$
– Ankit Kumar
Dec 26 '18 at 12:06
add a comment |
$begingroup$
Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
$$|MA|^2=|MB|^2+|MC|^2$$
$$implies y^2+2sqrt3ay-a^2=0$$
$$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative
Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)
$endgroup$
Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
$$|MA|^2=|MB|^2+|MC|^2$$
$$implies y^2+2sqrt3ay-a^2=0$$
$$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative
Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)
answered Dec 26 '18 at 11:57
Ankit KumarAnkit Kumar
1,494221
1,494221
$begingroup$
How do you get, $sqrt3 a$
$endgroup$
– Elementary
Dec 26 '18 at 12:05
$begingroup$
Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
$endgroup$
– Ankit Kumar
Dec 26 '18 at 12:06
add a comment |
$begingroup$
How do you get, $sqrt3 a$
$endgroup$
– Elementary
Dec 26 '18 at 12:05
$begingroup$
Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
$endgroup$
– Ankit Kumar
Dec 26 '18 at 12:06
$begingroup$
How do you get, $sqrt3 a$
$endgroup$
– Elementary
Dec 26 '18 at 12:05
$begingroup$
How do you get, $sqrt3 a$
$endgroup$
– Elementary
Dec 26 '18 at 12:05
$begingroup$
Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
$endgroup$
– Ankit Kumar
Dec 26 '18 at 12:06
$begingroup$
Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
$endgroup$
– Ankit Kumar
Dec 26 '18 at 12:06
add a comment |
$begingroup$
That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.
$endgroup$
1
$begingroup$
No, this is wrong approach.
$endgroup$
– Elementary
Dec 26 '18 at 11:51
1
$begingroup$
The answer is incorrect too
$endgroup$
– Ankit Kumar
Dec 26 '18 at 11:52
$begingroup$
But why doesn't that work?
$endgroup$
– Viktor Glombik
Dec 26 '18 at 12:39
add a comment |
$begingroup$
That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.
$endgroup$
1
$begingroup$
No, this is wrong approach.
$endgroup$
– Elementary
Dec 26 '18 at 11:51
1
$begingroup$
The answer is incorrect too
$endgroup$
– Ankit Kumar
Dec 26 '18 at 11:52
$begingroup$
But why doesn't that work?
$endgroup$
– Viktor Glombik
Dec 26 '18 at 12:39
add a comment |
$begingroup$
That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.
$endgroup$
That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.
answered Dec 26 '18 at 11:48
FrostFrost
1776
1776
1
$begingroup$
No, this is wrong approach.
$endgroup$
– Elementary
Dec 26 '18 at 11:51
1
$begingroup$
The answer is incorrect too
$endgroup$
– Ankit Kumar
Dec 26 '18 at 11:52
$begingroup$
But why doesn't that work?
$endgroup$
– Viktor Glombik
Dec 26 '18 at 12:39
add a comment |
1
$begingroup$
No, this is wrong approach.
$endgroup$
– Elementary
Dec 26 '18 at 11:51
1
$begingroup$
The answer is incorrect too
$endgroup$
– Ankit Kumar
Dec 26 '18 at 11:52
$begingroup$
But why doesn't that work?
$endgroup$
– Viktor Glombik
Dec 26 '18 at 12:39
1
1
$begingroup$
No, this is wrong approach.
$endgroup$
– Elementary
Dec 26 '18 at 11:51
$begingroup$
No, this is wrong approach.
$endgroup$
– Elementary
Dec 26 '18 at 11:51
1
1
$begingroup$
The answer is incorrect too
$endgroup$
– Ankit Kumar
Dec 26 '18 at 11:52
$begingroup$
The answer is incorrect too
$endgroup$
– Ankit Kumar
Dec 26 '18 at 11:52
$begingroup$
But why doesn't that work?
$endgroup$
– Viktor Glombik
Dec 26 '18 at 12:39
$begingroup$
But why doesn't that work?
$endgroup$
– Viktor Glombik
Dec 26 '18 at 12:39
add a comment |