Prove that in similar triangles ratio of correspondent medians is same as ratio of correspondent sides
up vote
2
down vote
favorite
I had a math exam today about geometry and similar triangles.
One of our math puzzle wanted us to proves something. Now I’ll explain that for you and if you help me I won’t lose 2 points of my midterm exam! So imagine that I’m your student and you’ve asked this question and I’ve answered like that.
QUESTION: We have two similar triangles. Prove that ratio of correspondent medians is same as ratio of correspondent sides.
MY ANSWER: We suppose two similar triangles, $triangle ABC$ and $triangle A’B’C’$. and also I did not mention that sides are equal! I mean $AB neq A’B’$ , $AC neq A’C’$ , $BC neq B’C’$.
And then I draw diagram 2. You can take a look here.
Actually I combined shapes in diagram 1, and I just draw diagram 2 in my exam paper. (I changed name of points in diagrams to explain what I answered better)
I wrote that we know:
$$triangle ABCthicksim triangle AMN$$
$$MN parallel BC$$
$$BH=HC$$
$$MO=ON$$
$AO space, AH$ are medians
So I continued based on thales theorem:
$$frac{AM}{MB}=frac{AO}{OH}$$
$$frac{AN}{NC}=frac{AO}{OH}$$
Thus $$frac{AM}{MB}=frac{AN}{NC}$$
On the other side :
$$frac{AM}{MB}=frac{AN}{NC}=frac{AO}{OH}$$
And finally he gave me a big beautiful zero! I don’t know why and I hadn’t a change to talk to him. What’s your Idea? Is my answer OK? If yes tell me why. Because I’m going to convince him.
geometry euclidean-geometry
edited Nov 19 at 16:48
Micah
29.5k1363104
asked Nov 19 at 16:33
user602338
1537
add a comment |
up vote
2
down vote
favorite
I had a math exam today about geometry and similar triangles.
One of our math puzzle wanted us to proves something. Now I’ll explain that for you and if you help me I won’t lose 2 points of my midterm exam! So imagine that I’m your student and you’ve asked this question and I’ve answered like that.
QUESTION: We have two similar triangles. Prove that ratio of correspondent medians is same as ratio of correspondent sides.
MY ANSWER: We suppose two similar triangles, $triangle ABC$ and $triangle A’B’C’$. and also I did not mention that sides are equal! I mean $AB neq A’B’$ , $AC neq A’C’$ , $BC neq B’C’$.
And then I draw diagram 2. You can take a look here.
Actually I combined shapes in diagram 1, and I just draw diagram 2 in my exam paper. (I changed name of points in diagrams to explain what I answered better)
I wrote that we know:
$$triangle ABCthicksim triangle AMN$$
$$MN parallel BC$$
$$BH=HC$$
$$MO=ON$$
$AO space, AH$ are medians
So I continued based on thales theorem:
$$frac{AM}{MB}=frac{AO}{OH}$$
$$frac{AN}{NC}=frac{AO}{OH}$$
Thus $$frac{AM}{MB}=frac{AN}{NC}$$
On the other side :
$$frac{AM}{MB}=frac{AN}{NC}=frac{AO}{OH}$$
And finally he gave me a big beautiful zero! I don’t know why and I hadn’t a change to talk to him. What’s your Idea? Is my answer OK? If yes tell me why. Because I’m going to convince him.
geometry euclidean-geometry
edited Nov 19 at 16:48
Micah
29.5k1363104
asked Nov 19 at 16:33
user602338
1537
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I had a math exam today about geometry and similar triangles.
One of our math puzzle wanted us to proves something. Now I’ll explain that for you and if you help me I won’t lose 2 points of my midterm exam! So imagine that I’m your student and you’ve asked this question and I’ve answered like that.
QUESTION: We have two similar triangles. Prove that ratio of correspondent medians is same as ratio of correspondent sides.
MY ANSWER: We suppose two similar triangles, $triangle ABC$ and $triangle A’B’C’$. and also I did not mention that sides are equal! I mean $AB neq A’B’$ , $AC neq A’C’$ , $BC neq B’C’$.
And then I draw diagram 2. You can take a look here.
Actually I combined shapes in diagram 1, and I just draw diagram 2 in my exam paper. (I changed name of points in diagrams to explain what I answered better)
I wrote that we know:
$$triangle ABCthicksim triangle AMN$$
$$MN parallel BC$$
$$BH=HC$$
$$MO=ON$$
$AO space, AH$ are medians
So I continued based on thales theorem:
$$frac{AM}{MB}=frac{AO}{OH}$$
$$frac{AN}{NC}=frac{AO}{OH}$$
Thus $$frac{AM}{MB}=frac{AN}{NC}$$
On the other side :
$$frac{AM}{MB}=frac{AN}{NC}=frac{AO}{OH}$$
And finally he gave me a big beautiful zero! I don’t know why and I hadn’t a change to talk to him. What’s your Idea? Is my answer OK? If yes tell me why. Because I’m going to convince him.
geometry euclidean-geometry
edited Nov 19 at 16:48
Micah
29.5k1363104
asked Nov 19 at 16:33
user602338
1537
I had a math exam today about geometry and similar triangles.
One of our math puzzle wanted us to proves something. Now I’ll explain that for you and if you help me I won’t lose 2 points of my midterm exam! So imagine that I’m your student and you’ve asked this question and I’ve answered like that.
QUESTION: We have two similar triangles. Prove that ratio of correspondent medians is same as ratio of correspondent sides.
MY ANSWER: We suppose two similar triangles, $triangle ABC$ and $triangle A’B’C’$. and also I did not mention that sides are equal! I mean $AB neq A’B’$ , $AC neq A’C’$ , $BC neq B’C’$.
And then I draw diagram 2. You can take a look here.
Actually I combined shapes in diagram 1, and I just draw diagram 2 in my exam paper. (I changed name of points in diagrams to explain what I answered better)
I wrote that we know:
$$triangle ABCthicksim triangle AMN$$
$$MN parallel BC$$
$$BH=HC$$
$$MO=ON$$
$AO space, AH$ are medians
So I continued based on thales theorem:
$$frac{AM}{MB}=frac{AO}{OH}$$
$$frac{AN}{NC}=frac{AO}{OH}$$
Thus $$frac{AM}{MB}=frac{AN}{NC}$$
On the other side :
$$frac{AM}{MB}=frac{AN}{NC}=frac{AO}{OH}$$
And finally he gave me a big beautiful zero! I don’t know why and I hadn’t a change to talk to him. What’s your Idea? Is my answer OK? If yes tell me why. Because I’m going to convince him.
geometry euclidean-geometry
geometry euclidean-geometry
edited Nov 19 at 16:48
Micah
29.5k1363104
asked Nov 19 at 16:33
user602338
1537
edited Nov 19 at 16:48
Micah
29.5k1363104
asked Nov 19 at 16:33
user602338
1537
edited Nov 19 at 16:48
Micah
29.5k1363104
edited Nov 19 at 16:48
Micah
29.5k1363104
edited Nov 19 at 16:48
Micah
29.5k1363104
29.5k1363104
asked Nov 19 at 16:33
user602338
1537
asked Nov 19 at 16:33
user602338
1537
asked Nov 19 at 16:33
user602338
1537
1537
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
Staying with diagram 1, by similarity$$frac{AB}{BC}=frac{A'B'}{B'C'}$$But since $M$ and $M'$ are midpoints we know$$frac{AB}{BM}=frac{A'B'}{B'M'}$$And we're given $$angle B=angle B'$$Therefore (Euclid, Elements VI, 4 & 6)$$triangle ABMsimtriangle A'B'M'$$from which it follows that$$frac{AB}{A'B'}=frac{AM}{A'M'}$$
I'm not sure it helped to superimpose the triangles (diagram 2). Your concluding proportion has lines $MB$, $NC$, $OH$, which are differences of corresponding sides, but doesn't the question call for $AB$, $AC$, $AH$ instead? You're almost there, and the teacher's zero seems a bit harsh, but you may not win your case.
answered Nov 21 at 19:32
Edward Porcella
1,4111411
Yep! I agree with you Mr.Porcella. I actually hadn't any Ideas about AB , AC , and AH and still no! Because our teacher didn't prove that and when i talked to one of my classmates his idea was that.
– user602338
Nov 22 at 7:10
Mr Porcella you are a teacher. Actually you are a retired one! So as a student who is going to study a lot in chemistry, physics, biology, literature , english, arabic, geography and so on for the entrance exam to medical college, is it reasonable that I spare my time with these math proofs that does'nt really help me!?
– user602338
Nov 22 at 7:14
Well you've got a lot to study, but mathematical proofs can help train you in ways that might not be obvious at first. Someone asked Abraham Lincoln once how he learned to think and write so clearly, and he said it was by studying the proofs in the first six books of Euclid's Elements. All the best to you.
– Edward Porcella
Nov 22 at 16:18
Yeah surely I agree with you there! Good night!
– user602338
Nov 22 at 18:07
add a comment |
up vote
1
down vote
I think you have a reasonable idea here, but your proof is incomplete. In order to apply Thales' theorem in this way, you need to know that $A$, $O$, and $H$ are collinear, and you haven't given any reason why they should be.
Notice that you haven't ever used the fact that $O$ and $H$ are midpoints. This is what you will need to prove collinearity.
answered Nov 19 at 17:07
Micah
29.5k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
Nov 19 at 17:26
APPLY THALES' THEOREM AGAIN TO SHOW THAT THE INTERSECTION OF $overline{AH}$ WITH $overline{MN}$ IS THE MIDPOINT OF $overline{MN}$, AND THUS MUST COINCIDE WITH $O$. (also, stop shouting)
– Micah
Nov 19 at 17:29
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
Nov 19 at 17:39
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Staying with diagram 1, by similarity$$frac{AB}{BC}=frac{A'B'}{B'C'}$$But since $M$ and $M'$ are midpoints we know$$frac{AB}{BM}=frac{A'B'}{B'M'}$$And we're given $$angle B=angle B'$$Therefore (Euclid, Elements VI, 4 & 6)$$triangle ABMsimtriangle A'B'M'$$from which it follows that$$frac{AB}{A'B'}=frac{AM}{A'M'}$$
I'm not sure it helped to superimpose the triangles (diagram 2). Your concluding proportion has lines $MB$, $NC$, $OH$, which are differences of corresponding sides, but doesn't the question call for $AB$, $AC$, $AH$ instead? You're almost there, and the teacher's zero seems a bit harsh, but you may not win your case.
answered Nov 21 at 19:32
Edward Porcella
1,4111411
Yep! I agree with you Mr.Porcella. I actually hadn't any Ideas about AB , AC , and AH and still no! Because our teacher didn't prove that and when i talked to one of my classmates his idea was that.
– user602338
Nov 22 at 7:10
Mr Porcella you are a teacher. Actually you are a retired one! So as a student who is going to study a lot in chemistry, physics, biology, literature , english, arabic, geography and so on for the entrance exam to medical college, is it reasonable that I spare my time with these math proofs that does'nt really help me!?
– user602338
Nov 22 at 7:14
Well you've got a lot to study, but mathematical proofs can help train you in ways that might not be obvious at first. Someone asked Abraham Lincoln once how he learned to think and write so clearly, and he said it was by studying the proofs in the first six books of Euclid's Elements. All the best to you.
– Edward Porcella
Nov 22 at 16:18
Yeah surely I agree with you there! Good night!
– user602338
Nov 22 at 18:07
add a comment |
up vote
1
down vote
accepted
Staying with diagram 1, by similarity$$frac{AB}{BC}=frac{A'B'}{B'C'}$$But since $M$ and $M'$ are midpoints we know$$frac{AB}{BM}=frac{A'B'}{B'M'}$$And we're given $$angle B=angle B'$$Therefore (Euclid, Elements VI, 4 & 6)$$triangle ABMsimtriangle A'B'M'$$from which it follows that$$frac{AB}{A'B'}=frac{AM}{A'M'}$$
I'm not sure it helped to superimpose the triangles (diagram 2). Your concluding proportion has lines $MB$, $NC$, $OH$, which are differences of corresponding sides, but doesn't the question call for $AB$, $AC$, $AH$ instead? You're almost there, and the teacher's zero seems a bit harsh, but you may not win your case.
answered Nov 21 at 19:32
Edward Porcella
1,4111411
Yep! I agree with you Mr.Porcella. I actually hadn't any Ideas about AB , AC , and AH and still no! Because our teacher didn't prove that and when i talked to one of my classmates his idea was that.
– user602338
Nov 22 at 7:10
Mr Porcella you are a teacher. Actually you are a retired one! So as a student who is going to study a lot in chemistry, physics, biology, literature , english, arabic, geography and so on for the entrance exam to medical college, is it reasonable that I spare my time with these math proofs that does'nt really help me!?
– user602338
Nov 22 at 7:14
Well you've got a lot to study, but mathematical proofs can help train you in ways that might not be obvious at first. Someone asked Abraham Lincoln once how he learned to think and write so clearly, and he said it was by studying the proofs in the first six books of Euclid's Elements. All the best to you.
– Edward Porcella
Nov 22 at 16:18
Yeah surely I agree with you there! Good night!
– user602338
Nov 22 at 18:07
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Staying with diagram 1, by similarity$$frac{AB}{BC}=frac{A'B'}{B'C'}$$But since $M$ and $M'$ are midpoints we know$$frac{AB}{BM}=frac{A'B'}{B'M'}$$And we're given $$angle B=angle B'$$Therefore (Euclid, Elements VI, 4 & 6)$$triangle ABMsimtriangle A'B'M'$$from which it follows that$$frac{AB}{A'B'}=frac{AM}{A'M'}$$
I'm not sure it helped to superimpose the triangles (diagram 2). Your concluding proportion has lines $MB$, $NC$, $OH$, which are differences of corresponding sides, but doesn't the question call for $AB$, $AC$, $AH$ instead? You're almost there, and the teacher's zero seems a bit harsh, but you may not win your case.
answered Nov 21 at 19:32
Edward Porcella
1,4111411
Staying with diagram 1, by similarity$$frac{AB}{BC}=frac{A'B'}{B'C'}$$But since $M$ and $M'$ are midpoints we know$$frac{AB}{BM}=frac{A'B'}{B'M'}$$And we're given $$angle B=angle B'$$Therefore (Euclid, Elements VI, 4 & 6)$$triangle ABMsimtriangle A'B'M'$$from which it follows that$$frac{AB}{A'B'}=frac{AM}{A'M'}$$
I'm not sure it helped to superimpose the triangles (diagram 2). Your concluding proportion has lines $MB$, $NC$, $OH$, which are differences of corresponding sides, but doesn't the question call for $AB$, $AC$, $AH$ instead? You're almost there, and the teacher's zero seems a bit harsh, but you may not win your case.
answered Nov 21 at 19:32
Edward Porcella
1,4111411
answered Nov 21 at 19:32
Edward Porcella
1,4111411
answered Nov 21 at 19:32
Edward Porcella
1,4111411
answered Nov 21 at 19:32
Edward Porcella
1,4111411
1,4111411
Yep! I agree with you Mr.Porcella. I actually hadn't any Ideas about AB , AC , and AH and still no! Because our teacher didn't prove that and when i talked to one of my classmates his idea was that.
– user602338
Nov 22 at 7:10
Mr Porcella you are a teacher. Actually you are a retired one! So as a student who is going to study a lot in chemistry, physics, biology, literature , english, arabic, geography and so on for the entrance exam to medical college, is it reasonable that I spare my time with these math proofs that does'nt really help me!?
– user602338
Nov 22 at 7:14
Well you've got a lot to study, but mathematical proofs can help train you in ways that might not be obvious at first. Someone asked Abraham Lincoln once how he learned to think and write so clearly, and he said it was by studying the proofs in the first six books of Euclid's Elements. All the best to you.
– Edward Porcella
Nov 22 at 16:18
Yeah surely I agree with you there! Good night!
– user602338
Nov 22 at 18:07
add a comment |
Yep! I agree with you Mr.Porcella. I actually hadn't any Ideas about AB , AC , and AH and still no! Because our teacher didn't prove that and when i talked to one of my classmates his idea was that.
– user602338
Nov 22 at 7:10
Mr Porcella you are a teacher. Actually you are a retired one! So as a student who is going to study a lot in chemistry, physics, biology, literature , english, arabic, geography and so on for the entrance exam to medical college, is it reasonable that I spare my time with these math proofs that does'nt really help me!?
– user602338
Nov 22 at 7:14
Well you've got a lot to study, but mathematical proofs can help train you in ways that might not be obvious at first. Someone asked Abraham Lincoln once how he learned to think and write so clearly, and he said it was by studying the proofs in the first six books of Euclid's Elements. All the best to you.
– Edward Porcella
Nov 22 at 16:18
Yeah surely I agree with you there! Good night!
– user602338
Nov 22 at 18:07
Yep! I agree with you Mr.Porcella. I actually hadn't any Ideas about AB , AC , and AH and still no! Because our teacher didn't prove that and when i talked to one of my classmates his idea was that.
– user602338
Nov 22 at 7:10
Yep! I agree with you Mr.Porcella. I actually hadn't any Ideas about AB , AC , and AH and still no! Because our teacher didn't prove that and when i talked to one of my classmates his idea was that.
– user602338
Nov 22 at 7:10
Mr Porcella you are a teacher. Actually you are a retired one! So as a student who is going to study a lot in chemistry, physics, biology, literature , english, arabic, geography and so on for the entrance exam to medical college, is it reasonable that I spare my time with these math proofs that does'nt really help me!?
– user602338
Nov 22 at 7:14
Mr Porcella you are a teacher. Actually you are a retired one! So as a student who is going to study a lot in chemistry, physics, biology, literature , english, arabic, geography and so on for the entrance exam to medical college, is it reasonable that I spare my time with these math proofs that does'nt really help me!?
– user602338
Nov 22 at 7:14
Well you've got a lot to study, but mathematical proofs can help train you in ways that might not be obvious at first. Someone asked Abraham Lincoln once how he learned to think and write so clearly, and he said it was by studying the proofs in the first six books of Euclid's Elements. All the best to you.
– Edward Porcella
Nov 22 at 16:18
Well you've got a lot to study, but mathematical proofs can help train you in ways that might not be obvious at first. Someone asked Abraham Lincoln once how he learned to think and write so clearly, and he said it was by studying the proofs in the first six books of Euclid's Elements. All the best to you.
– Edward Porcella
Nov 22 at 16:18
Yeah surely I agree with you there! Good night!
– user602338
Nov 22 at 18:07
Yeah surely I agree with you there! Good night!
– user602338
Nov 22 at 18:07
add a comment |
up vote
1
down vote
I think you have a reasonable idea here, but your proof is incomplete. In order to apply Thales' theorem in this way, you need to know that $A$, $O$, and $H$ are collinear, and you haven't given any reason why they should be.
Notice that you haven't ever used the fact that $O$ and $H$ are midpoints. This is what you will need to prove collinearity.
answered Nov 19 at 17:07
Micah
29.5k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
Nov 19 at 17:26
APPLY THALES' THEOREM AGAIN TO SHOW THAT THE INTERSECTION OF $overline{AH}$ WITH $overline{MN}$ IS THE MIDPOINT OF $overline{MN}$, AND THUS MUST COINCIDE WITH $O$. (also, stop shouting)
– Micah
Nov 19 at 17:29
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
Nov 19 at 17:39
add a comment |
up vote
1
down vote
I think you have a reasonable idea here, but your proof is incomplete. In order to apply Thales' theorem in this way, you need to know that $A$, $O$, and $H$ are collinear, and you haven't given any reason why they should be.
Notice that you haven't ever used the fact that $O$ and $H$ are midpoints. This is what you will need to prove collinearity.
answered Nov 19 at 17:07
Micah
29.5k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
Nov 19 at 17:26
APPLY THALES' THEOREM AGAIN TO SHOW THAT THE INTERSECTION OF $overline{AH}$ WITH $overline{MN}$ IS THE MIDPOINT OF $overline{MN}$, AND THUS MUST COINCIDE WITH $O$. (also, stop shouting)
– Micah
Nov 19 at 17:29
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
Nov 19 at 17:39
add a comment |
up vote
1
down vote
up vote
1
down vote
I think you have a reasonable idea here, but your proof is incomplete. In order to apply Thales' theorem in this way, you need to know that $A$, $O$, and $H$ are collinear, and you haven't given any reason why they should be.
Notice that you haven't ever used the fact that $O$ and $H$ are midpoints. This is what you will need to prove collinearity.
answered Nov 19 at 17:07
Micah
29.5k1363104
I think you have a reasonable idea here, but your proof is incomplete. In order to apply Thales' theorem in this way, you need to know that $A$, $O$, and $H$ are collinear, and you haven't given any reason why they should be.
Notice that you haven't ever used the fact that $O$ and $H$ are midpoints. This is what you will need to prove collinearity.
answered Nov 19 at 17:07
Micah
29.5k1363104
answered Nov 19 at 17:07
Micah
29.5k1363104
answered Nov 19 at 17:07
Micah
29.5k1363104
answered Nov 19 at 17:07
Micah
29.5k1363104
29.5k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
Nov 19 at 17:26
APPLY THALES' THEOREM AGAIN TO SHOW THAT THE INTERSECTION OF $overline{AH}$ WITH $overline{MN}$ IS THE MIDPOINT OF $overline{MN}$, AND THUS MUST COINCIDE WITH $O$. (also, stop shouting)
– Micah
Nov 19 at 17:29
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
Nov 19 at 17:39
add a comment |
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
Nov 19 at 17:26
APPLY THALES' THEOREM AGAIN TO SHOW THAT THE INTERSECTION OF $overline{AH}$ WITH $overline{MN}$ IS THE MIDPOINT OF $overline{MN}$, AND THUS MUST COINCIDE WITH $O$. (also, stop shouting)
– Micah
Nov 19 at 17:29
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
Nov 19 at 17:39
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
Nov 19 at 17:26
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
Nov 19 at 17:26
APPLY THALES' THEOREM AGAIN TO SHOW THAT THE INTERSECTION OF $overline{AH}$ WITH $overline{MN}$ IS THE MIDPOINT OF $overline{MN}$, AND THUS MUST COINCIDE WITH $O$. (also, stop shouting)
– Micah
Nov 19 at 17:29
APPLY THALES' THEOREM AGAIN TO SHOW THAT THE INTERSECTION OF $overline{AH}$ WITH $overline{MN}$ IS THE MIDPOINT OF $overline{MN}$, AND THUS MUST COINCIDE WITH $O$. (also, stop shouting)
– Micah
Nov 19 at 17:29
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
Nov 19 at 17:39
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
Nov 19 at 17:39
add a comment |
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