Problem using triangular inequality

[CarlosP]

You wrote: "Then we need find the closest possible point A such that |AP| is the minimum distance"
So, where should we put A to minimize |AP|?
This would give us a potential solution.

We should put A in the point O

 

[lev888]

We should put A in the point O

Yes. B will also coincide with O.
Now we can try proving that moving A and B away from O increases |BQ| + 2 |AP|. I don't yet see how this can be done.

 

[Dr.Peterson]

Just to let you know, I've been playing with the problem by modeling it in GeoGebra, and the solution (which I haven't found yet apart from manually moving points around to find a minimum) will not be nearly as simple as A=B=O.

@CarlosP, can you tell us where this problem came from, and what the context suggests about appropriate methods? You've indicated that it is expected to use the triangle inequality, but do you know that for sure? Might you need to use coordinate geometry or calculus? Or are geometrical transformations like your use of reflections certain to be the way to go? (How about dilations?)

 

[CarlosP]

Just to let you know, I've been playing with the problem by modeling it in GeoGebra, and the solution (which I haven't found yet apart from manually moving points around to find a minimum) will not be nearly as simple as A=B=O.

@CarlosP, can you tell us where this problem came from, and what the context suggests about appropriate methods? You've indicated that it is expected to use the triangle inequality, but do you know that for sure? Might you need to use coordinate geometry or calculus? Or are geometrical transformations like your use of reflections certain to be the way to go? (How about dilations?)

This problem is from exam of geometry that i have and i want solve this problem. My teacher solve problems easier then that using triangle inequality and reflections only. So i thought in this exam is the same but much harder. This is the problem. It is in portuguese

 

[Dr.Peterson]

Although I doubt it matters, could you give us a readable copy of the problem, so we can make sure nothing has been missed? It still seems like a much more difficult problem than you are expecting it to be, so I'm hoping you translated something wrong.

Are you saying this is on an actual exam that you are taking, or is it just a practice exam?

It's helpful to know this a geometry exam. What other transformations have you used besides reflection? Is dilation (stretching) included?

 

[pka]

Although I doubt it matters, could you give us a readable copy of the problem, so we can make sure nothing has been missed? It still seems like a much more difficult problem than you are expecting it to be, so I'm hoping you translated something wrong. Are you saying this is on an actual exam that you are taking, or is it just a practice exam? It's helpful to know this a geometry exam. What other transformations have you used besides reflection? Is dilation (stretching) included?

Of all the poorly written questions we have had lately, this in the worst. I have not even a clue as to what it could mean.

 

[Dr.Peterson]

The problem as first stated is translated from the Portuguese in the (largely unreadable) image into mostly reasonable English, so I'd say it's not bad. I just want to clarify some details in case they make the problem easier. But here's my edited version of the translation we were given:

Let l be a line, O a point on l, and P and Q two points not on l, located in the same semiplane bounded by l (that is, on the same side of l). Determine from among all pairs of points A, B on line l such that O is between A and B and | OB | = 2 | OA |, the pair for which the expression | BQ | + 2 | AP | takes the maximum value.​

The hard part is solving it!

 

[lev888]

The problem as first stated is translated from the Portuguese in the (largely unreadable) image into mostly reasonable English, so I'd say it's not bad. I just want to clarify some details in case they make the problem easier. But here's my edited version of the translation we were given:

Let l be a line, O a point on l, and P and Q two points not on l, located in the same semiplane bounded by l (that is, on the same side of l). Determine from among all pairs of points A, B on line l such that O is between A and B and | OB | = 2 | OA |, the pair for which the expression | BQ | + 2 | AP | takes the maximum value.​

The hard part is solving it!

Should be minimum.

 

[CarlosP]

The problem as first stated is translated from the Portuguese in the (largely unreadable) image into mostly reasonable English, so I'd say it's not bad. I just want to clarify some details in case they make the problem easier. But here's my edited version of the translation we were given:

Let l be a line, O a point on l, and P and Q two points not on l, located in the same semiplane bounded by l (that is, on the same side of l). Determine from among all pairs of points A, B on line l such that O is between A and B and | OB | = 2 | OA |, the pair for which the expression | BQ | + 2 | AP | takes the maximum value.​

The hard part is solving it!

we want | BQ | + 2 | AP | takes the minimum value.

 

[CarlosP]

Although I doubt it matters, could you give us a readable copy of the problem, so we can make sure nothing has been missed? It still seems like a much more difficult problem than you are expecting it to be, so I'm hoping you translated something wrong.

Are you saying this is on an actual exam that you are taking, or is it just a practice exam?

It's helpful to know this a geometry exam. What other transformations have you used besides reflection? Is dilation (stretching) included?

this question came out last year's exam from geometry given by my teacher. We only learn 4 transformations : translation, reflection, rotation and sliding reflections. If you want i can send you the program of my discipline.

 

[CarlosP]

The problem as first stated is translated from the Portuguese in the (largely unreadable) image into mostly reasonable English, so I'd say it's not bad. I just want to clarify some details in case they make the problem easier. But here's my edited version of the translation we were given:

Let l be a line, O a point on l, and P and Q two points not on l, located in the same semiplane bounded by l (that is, on the same side of l). Determine from among all pairs of points A, B on line l such that O is between A and B and | OB | = 2 | OA |, the pair for which the expression | BQ | + 2 | AP | takes the maximum value.​

The hard part is solving it!

this is the problem in portuguese https://prnt.sc/ottiwj .

 

[Dr.Peterson]

Thanks. I didn't think your translation missed anything, but I wanted to make sure. Google translates it as

Let l be a line, O be a point of l, and P, Q two points outside l located in the same semiplane bounded by l. Determine from all pairs of points A, B ∈ l such that O ∈ AB and | OB | = 2 | OA |, the one for which the expression | BQ | + 2 | AP | takes the minimum value.​

Apart from typing minimum as maximum, your translation was fine, and I see nothing hidden that changes my understanding of it.

I'll continue looking; I realized last night that I had misread one detail that may suggest different things to try. Since you have studied only isometries (distance-preserving transformations associated with congruence), and not dilations (which are related to similarities), I won't consider using the latter, but it's possible that similar triangles will be involved (related to the multiplications by 2).

 

[CarlosP]

Thanks. I didn't think your translation missed anything, but I wanted to make sure. Google translates it as

Let l be a line, O be a point of l, and P, Q two points outside l located in the same semiplane bounded by l. Determine from all pairs of points A, B ∈ l such that O ∈ AB and | OB | = 2 | OA |, the one for which the expression | BQ | + 2 | AP | takes the minimum value.​

Apart from typing minimum as maximum, your translation was fine, and I see nothing hidden that changes my understanding of it.

I'll continue looking; I realized last night that I had misread one detail that may suggest different things to try. Since you have studied only isometries (distance-preserving transformations associated with congruence), and not dilations (which are related to similarities), I won't consider using the latter, but it's possible that similar triangles will be involved (related to the multiplications by 2).

Thanks for the help

 

[Dr.Peterson]

I've figured it out; I was hindered by trying to maximize |AP| + 2|BQ| instead of |BQ| + 2|AP|.

See if this figure helps:

 

[CarlosP]

I've figured it out; I was hindered by trying to maximize |AP| + 2|BQ| instead of |BQ| + 2|AP|.

See if this figure helps:
View attachment 13292

Using the similarity of triangles we see that to minimize | BQ | + 2 | AP | is the same as minimizing | A'Q '| + | A'P'|. Let A'P'Q' be a triangle. So to minimize | A'Q' | + | A'P'| , A' must belong to line segment P'Q'. Then we find A', which reflecting on line l gives point B. Then determining the midpoint of A'O we find point A.

 

[Dr.Peterson]

Good. There's probably more to say in order to give a full answer (e.g. stating how P' and Q' are found), and there may be a simpler way to state the final answer, but you've got it.

It wasn't nearly as hard as it seemed for a while! And I should have seen it much sooner, if I'd been thinking right.

By the way, there's another English language issue to mention. When you say "reflecting on line l", it confuses me, because I don't think we say that here; it sounds like "reflecting in line l", which is another way to say "reflecting across, or over, or through, line l". I would say "reflecting A' in [or through] point O". But I'm sure your terminology is correct in your language and in your curriculum (though I would think some mention needs to be made of point O).

 

[CarlosP]

Good. There's probably more to say in order to give a full answer (e.g. stating how P' and Q' are found), and there may be a simpler way to state the final answer, but you've got it.

It wasn't nearly as hard as it seemed for a while! And I should have seen it much sooner, if I'd been thinking right.

By the way, there's another English language issue to mention. When you say "reflecting on line l", it confuses me, because I don't think we say that here; it sounds like "reflecting in line l", which is another way to say "reflecting across, or over, or through, line l". I would say "reflecting A' in [or through] point O". But I'm sure your terminology is correct in your language and in your curriculum (though I would think some mention needs to be made of point O).

Thank you for your vision and that clue yes you're right i should give a full answer by saying how i found that points P' and Q'.
I would say "reflecting A in point O", seems better to me