Mixing
Prove that in similar triangles ratio of correspondent medians is same as ratio of correspondent sides
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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 yesterday
Micah
29.4k1363104
asked yesterday
user602338
1256
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 yesterday
Micah
29.4k1363104
asked yesterday
user602338
1256
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 yesterday
Micah
29.4k1363104
asked yesterday
user602338
1256
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 yesterday
Micah
29.4k1363104
asked yesterday
user602338
1256
edited yesterday
Micah
29.4k1363104
asked yesterday
user602338
1256
edited yesterday
Micah
29.4k1363104
edited yesterday
Micah
29.4k1363104
edited yesterday
Micah
29.4k1363104
29.4k1363104
asked yesterday
user602338
1256
asked yesterday
user602338
1256
asked yesterday
user602338
1256
1256
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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 yesterday
Micah
29.4k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
yesterday
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
yesterday
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
yesterday
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 yesterday
Micah
29.4k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
yesterday
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
yesterday
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
yesterday
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 yesterday
Micah
29.4k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
yesterday
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
yesterday
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
yesterday
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 yesterday
Micah
29.4k1363104
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 yesterday
Micah
29.4k1363104
answered yesterday
Micah
29.4k1363104
answered yesterday
Micah
29.4k1363104
answered yesterday
Micah
29.4k1363104
29.4k1363104
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
yesterday
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
yesterday
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
yesterday
add a comment |
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
yesterday
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
yesterday
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
yesterday
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
yesterday
SO WOULD YOU TELL ME HOW CAN WE PROVE TGAT THEY ARE COLLINEAR?
– user602338
yesterday
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
yesterday
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
yesterday
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
yesterday
Oh sorry i did't shout! My keyboard capslock was on! Also im not a native formal speaker bro hh! :D
– user602338
yesterday
add a comment |
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