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Math Expert V
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Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Two Tangent circles A and B have radii r1 and r2, respectively. What is the distance between their centers?

(1) r1 = 3 inches
(2) r2 = 5 inches

Kudos for a correct solution.

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Math Expert V
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Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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3
E....
even when two radiis are given ,the two circles can be tangential on the exterior or one inside the other and thus interior, a situation on which ans will depend
Attachments tangents...png [ 7.3 KiB | Viewed 5978 times ]

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Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Bunuel wrote:
Two Tangent circles A and B have radii r1 and r2, respectively. What is the distance between their centers?

(1) r1 = 3 inches
(2) r2 = 5 inches

Kudos for a correct solution.

A great question that leads us to believe that the answer would be C (with the two radii lengths; however the tangent circles could be tangent on the exterior or the smaller one could be inside the larger one and therefore tangent on the interior. We get an answer of 8 in the first scenario and 2 in the second. Therefore, answer choice is E.
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Joined: 05 Mar 2015
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Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Bunuel wrote:
Two Tangent circles A and B have radii r1 and r2, respectively. What is the distance between their centers?

(1) r1 = 3 inches
(2) r2 = 5 inches

Kudos for a correct solution.

consider two cases as per fig below

One has difference of 2 units
And another has 8 units
Neither suff..

Ans E
Attachments 111.png [ 9.33 KiB | Viewed 8304 times ]

Intern  Joined: 17 Jul 2017
Posts: 15
Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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rohit8865 wrote:
Bunuel wrote:
Two Tangent circles A and B have radii r1 and r2, respectively. What is the distance between their centers?

(1) r1 = 3 inches
(2) r2 = 5 inches

Kudos for a correct solution.

consider two cases as per fig below

One has difference of 2 units
And another has 8 units
Neither suff..

Ans E

Hmm thanks but is this wrong? I don't understand why they have to be tangent like the way you showed and not like this
Attachments tangent question.png [ 7.18 KiB | Viewed 7447 times ]

Manager  B
Joined: 31 Dec 2016
Posts: 65
Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Samuelboyle96 wrote:
rohit8865 wrote:
Bunuel wrote:
Two Tangent circles A and B have radii r1 and r2, respectively. What is the distance between their centers?

(1) r1 = 3 inches
(2) r2 = 5 inches

Kudos for a correct solution.

consider two cases as per fig below

One has difference of 2 units
And another has 8 units
Neither suff..

Ans E

Hmm thanks but is this wrong? I don't understand why they have to be tangent like the way you showed and not like this

I guess no one is going to answer because obviously the GMAT makes up their definition of tangent circles. http://mathworld.wolfram.com/TangentCircles.html I have yet to find anywhere on the internet besides the GMAT define tangent circles as 2 circles where their centers line up perfectly.

See it is a little weird the GMAT mentions tangent circles lining up horizontally here but not in other places https://gmatclub.com/forum/the-figure-a ... 91244.html
Attachments tangent circles.png [ 15.28 KiB | Viewed 7327 times ]

GMAT Tutor G
Joined: 24 Jun 2008
Posts: 1806
Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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SamBoyle wrote:

See it is a little weird the GMAT mentions tangent circles lining up horizontally here but not in other places

Two distinct circles are tangent to each other if they touch in exactly one point. You posted a diagram two posts above of two circles that do not touch anywhere, but which share a tangent line. Those circles are not tangent circles. The concept of tangency has nothing to do with whether things 'line up horizontally', so if the diagrams you've seen have displayed tangent circles with centers along a horizontal line, that's likely only because it's easier to draw diagrams that way.

SamBoyle wrote:

I guess no one is going to answer because obviously the GMAT makes up their definition of tangent circles.

All of the math on the GMAT is the same as the math you'd learn anywhere else, so there is no math on the test that the GMAT "makes up".
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Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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SamBoyle wrote:
Samuelboyle96 wrote:
rohit8865 wrote:

consider two cases as per fig below

One has difference of 2 units
And another has 8 units
Neither suff..

Ans E

Hmm thanks but is this wrong? I don't understand why they have to be tangent like the way you showed and not like this

I guess no one is going to answer because obviously the GMAT makes up their definition of tangent circles. http://mathworld.wolfram.com/TangentCircles.html I have yet to find anywhere on the internet besides the GMAT define tangent circles as 2 circles where their centers line up perfectly.

See it is a little weird the GMAT mentions tangent circles lining up horizontally here but not in other places https://gmatclub.com/forum/the-figure-a ... 91244.html

I agree with what IanStewart has mentioned above. Additionally, there is no need to discuss this further as GMAT or any math book will define tangency as 1 point of contact. But do understand that GMAT is a timed exam and hence, you should stop working on a DS question as soon as you find 2 contradictory scenarios; you are not getting extra points for coming with 100 different scenarios to answer the same set of DS statements. Additionally, it is easier to visualize 'horizontal scenarios' out of many scenarios that negate the statements mentioned in this DS question. Not seeing something on the internet does not mean GMAT 'makes up' math concepts.

Hope this helps.
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Joined: 31 Dec 2016
Posts: 65
Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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IanStewart wrote:
SamBoyle wrote:

See it is a little weird the GMAT mentions tangent circles lining up horizontally here but not in other places

Two distinct circles are tangent to each other if they touch in exactly one point. You posted a diagram two posts above of two circles that do not touch anywhere, but which share a tangent line. Those circles are not tangent circles. The concept of tangency has nothing to do with whether things 'line up horizontally', so if the diagrams you've seen have displayed tangent circles with centers along a horizontal line, that's likely only because it's easier to draw diagrams that way.

SamBoyle wrote:

I guess no one is going to answer because obviously the GMAT makes up their definition of tangent circles.

All of the math on the GMAT is the same as the math you'd learn anywhere else, so there is no math on the test that the GMAT "makes up".

You just made a bald face lie or are perhaps blind if you think my circles aren't touching.

I figured out my answer. The way I drew the diagram my answer is correct. However you can draw the radius to the center of the circle anyway you want. So I could have drawn it by a shorter route. Hence my distance is correct but not the shortest possible distance
Attachments debate.png [ 11.25 KiB | Viewed 7214 times ]

Manager  B
Joined: 31 Dec 2016
Posts: 65
Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Engr2012 wrote:
SamBoyle wrote:

I agree with what IanStewart has mentioned above. Additionally, there is no need to discuss this further as GMAT or any math book will define tangency as 1 point of contact. But do understand that GMAT is a timed exam and hence, you should stop working on a DS question as soon as you find 2 contradictory scenarios; you are not getting extra points for coming with 100 different scenarios to answer the same set of DS statements. Additionally, it is easier to visualize 'horizontal scenarios' out of many scenarios that negate the statements mentioned in this DS question. Not seeing something on the internet does not mean GMAT 'makes up' math concepts.

Hope this helps.

No that does the opposite of helping anyone and is the worst possible advice. That response makes very little sense. You think on the test, I would come up with 100 different scenarios on a single question. or do you think it's important to understand how tangency works before the test? Very unproductive advice.

We are here to learn. Learning and knowledge leads to speed. Speed without knowledge is pointless. It would just be guessing
GMAT Tutor G
Joined: 24 Jun 2008
Posts: 1806
Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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2
SamBoyle wrote:
You just made a bald face lie or are perhaps blind if you think my circles aren't touching.

I figured out my answer. The way I drew the diagram my answer is correct. However you can draw the radius to the center of the circle anyway you want. So I could have drawn it by a shorter route. Hence my distance is correct but not the shortest possible distance

You drew a diagram of one circle with radius 5, and one circle with radius 3, where the distance between the centers of the two circles was √68. It is mathematically impossible for those two circles to touch; they're too far apart. If you diagram that situation, and in your diagram the circles appear to touch, then you've drawn the diagram incorrectly.

The rest of your reply is not correct, because you are not using the correct definition of the word "distance". Absent any qualifications, the "distance" between two points is always the straight line distance between them, which is the shortest possible distance. Your distance cannot be correct and not be the shortest possible distance.

If you continue to reply to people the way you have here, I think you'll discover that eventually other members will become disinclined to try to help you.
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Intern  Joined: 17 Jul 2017
Posts: 15
Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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IanStewart wrote:
SamBoyle wrote:
You just made a bald face lie or are perhaps blind if you think my circles aren't touching.

I figured out my answer. The way I drew the diagram my answer is correct. However you can draw the radius to the center of the circle anyway you want. So I could have drawn it by a shorter route. Hence my distance is correct but not the shortest possible distance

You drew a diagram of one circle with radius 5, and one circle with radius 3, where the distance between the centers of the two circles was √68. It is mathematically impossible for those two circles to touch; they're too far apart. If you diagram that situation, and in your diagram the circles appear to touch, then you've drawn the diagram incorrectly.

The rest of your reply is not correct, because you are not using the correct definition of the word "distance". Absent any qualifications, the "distance" between two points is always the straight line distance between them, which is the shortest possible distance. Your distance cannot be correct and not be the shortest possible distance.

If you continue to reply to people the way you have here, I think you'll discover that eventually other members will become disinclined to try to help you.

Thanks, ironically the best response I have gotten here though I don't ask many questions. I mostly just provide solutions to help other people out--I find the solutions here are often lacking--and assume you know things like external angle theorem and just list it and don't explain how it works. That's fine but I try to explain things like that when I solve. I do like to understand the nuances of the problem, so that I can get another different question more easily
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Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Samuelboyle96 wrote:
IanStewart wrote:
SamBoyle wrote:
You just made a bald face lie or are perhaps blind if you think my circles aren't touching.

I figured out my answer. The way I drew the diagram my answer is correct. However you can draw the radius to the center of the circle anyway you want. So I could have drawn it by a shorter route. Hence my distance is correct but not the shortest possible distance

You drew a diagram of one circle with radius 5, and one circle with radius 3, where the distance between the centers of the two circles was √68. It is mathematically impossible for those two circles to touch; they're too far apart. If you diagram that situation, and in your diagram the circles appear to touch, then you've drawn the diagram incorrectly.

The rest of your reply is not correct, because you are not using the correct definition of the word "distance". Absent any qualifications, the "distance" between two points is always the straight line distance between them, which is the shortest possible distance. Your distance cannot be correct and not be the shortest possible distance.

If you continue to reply to people the way you have here, I think you'll discover that eventually other members will become disinclined to try to help you.

Thanks, ironically the best response I have gotten here though I don't ask many questions. I mostly just provide solutions to help other people out--I find the solutions here are often lacking--and assume you know things like external angle theorem and just list it and don't explain how it works. That's fine but I try to explain things like that when I solve. I do like to understand the nuances of the problem, so that I can get another different question more easily

You cant expect answers complete in 'all' respects as there are n different things that need to be captured every single time. Experts on this forum are more than happy to answer your questions but only if you ask them properly. If you have an issue with some explanation; dont be condescending to the OP but add to the discussion by asking "hey you know what, I think you are assuming that people would know xyz. So, if you could explain how you came about determining the angle measure, that would be great". If you ask a question in this manner, there will be tons of experts who would be more than happy to help you out.

Another aspect to covering all possible concepts is that some questions are already at a slightly higher level of usual comprehension, e.g. probability questions with multiple issues to keep a track of. So without the OP mentioning what he has/has not done; no one would know where to start with their explanation and everyone will take the path of least resistance and mention the solution directly.

If you dont ask 'many questions', good for you. Either you know your concepts or are too afraid to get answers by posting your queries online.

TL;DR: there is a proper way of engaging in these forums. Stick to that method and there will be a lot of people willing to help you out. Stay humble and eager to learn.
Intern  Joined: 17 Jul 2017
Posts: 15
Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Engr2012 wrote:

You cant expect answers complete in 'all' respects as there are n different things that need to be captured every single time. Experts on this forum are more than happy to answer your questions but only if you ask them properly. If you have an issue with some explanation; dont be condescending to the OP but add to the discussion by asking "hey you know what, I think you are assuming that people would know xyz. So, if you could explain how you came about determining the angle measure, that would be great". If you ask a question in this manner, there will be tons of experts who would be more than happy to help you out.

Another aspect to covering all possible concepts is that some questions are already at a slightly higher level of usual comprehension, e.g. probability questions with multiple issues to keep a track of. So without the OP mentioning what he has/has not done; no one would know where to start with their explanation and everyone will take the path of least resistance and mention the solution directly.

If you dont ask 'many questions', good for you. Either you know your concepts or are too afraid to get answers by posting your queries online.

TL;DR: there is a proper way of engaging in these forums. Stick to that method and there will be a lot of people willing to help you out. Stay humble and eager to learn.

I think, I did highlight a very important concept. That circle can be tangent and not be touching horizontally. My point is while this seems so obvious to me now but I couldn't find anything on the internet showing 2 tangent circles and distance. Everyone else calculated 2 tangent circles on a line and the distance between.

I think my exploration of the concept should help more people better picture why the distance between 2 tangent circles on the outside is just their radius.
Attachments radius tangent circles.png [ 92.21 KiB | Viewed 7016 times ]

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Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Samuelboyle96 wrote:
I think, I did highlight a very important concept. That circle can be tangent and not be touching horizontally. My point is while this seems so obvious to me now but I couldn't find anything on the internet showing 2 tangent circles and distance. Everyone else calculated 2 tangent circles on a line and the distance between.

I think my exploration of the concept should help more people better picture why the distance between 2 tangent circles on the outside is just their radius.

You raise a very small but important concept for complete understanding of when multiple circles that are tangent to each other or touch each other at 1 and only 1 point. I understood your question the first time around and so did IanStewart .

For the sake of this question and for similar questions:

r1, r2 = radii of the 2 circles; where r2 > r1

For 2 circles touching each other:
a) If tangency is on the outside: distance between the centers = r1 + r2
b) If tangency is on the inside: distance between the centers = r2-r1

This relatively straightforward concept has much broader application:

a) If you are given that distance between the centers of any 2 circles = r1+r2 --> Circles are tangent on the outside.
b) If you are given that distance between the centers of any 2 circles = r2-r1 --> Circles are tangent on the inside.
c) If you are given that distance between the centers of any 2 circles > r2+r1 --> Circles are such that one is NOT inside the other and that they are NOT touching each other.
d) If you are given that distance between the centers of any 2 circles < r2+r1 --> Circles are intersect each other at 2 points
e) If you are given that distance between the centers of any 2 circles < r2-r1 --> One circle is inside the other and not touching.

Again, think about these intuitively instead of memorizing these observations. Important point is that these 'observations' have been derived from your understanding of how the distance between the centers of 2 circle varies with certain combinations of respective radii.

Finally, to your point of tangent circles might not be horizontal, that is a completely correct statement. It just so happens that there can be infinite points of tangency between 2 circles and if I give you a choice of showing this tangency, you will always/instinctly choose the horizontal scenario as it is more straightforward to visualize.

Hope this helps.
Intern  Joined: 17 Jul 2017
Posts: 15
Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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Engr2012 wrote:
Samuelboyle96 wrote:
I think, I did highlight a very important concept. That circle can be tangent and not be touching horizontally. My point is while this seems so obvious to me now but I couldn't find anything on the internet showing 2 tangent circles and distance. Everyone else calculated 2 tangent circles on a line and the distance between.

I think my exploration of the concept should help more people better picture why the distance between 2 tangent circles on the outside is just their radius.

You raise a very small but important concept for complete understanding of when multiple circles that are tangent to each other or touch each other at 1 and only 1 point. I understood your question the first time around and so did IanStewart .

For the sake of this question and for similar questions:

r1, r2 = radii of the 2 circles; where r2 > r1

For 2 circles touching each other:
a) If tangency is on the outside: distance between the centers = r1 + r2
b) If tangency is on the inside: distance between the centers = r2-r1

This relatively straightforward concept has much broader application:

a) If you are given that distance between the centers of any 2 circles = r1+r2 --> Circles are tangent on the outside.
b) If you are given that distance between the centers of any 2 circles = r2-r1 --> Circles are tangent on the inside.
c) If you are given that distance between the centers of any 2 circles > r2+r1 --> Circles are such that one is NOT inside the other and that they are NOT touching each other.
d) If you are given that distance between the centers of any 2 circles < r2+r1 --> Circles are intersect each other at 2 points
e) If you are given that distance between the centers of any 2 circles < r2-r1 --> One circle is inside the other and not touching.

Again, think about these intuitively instead of memorizing these observations. Important point is that these 'observations' have been derived from your understanding of how the distance between the centers of 2 circle varies with certain combinations of respective radii.

Finally, to your point of tangent circles might not be horizontal, that is a completely correct statement. It just so happens that there can be infinite points of tangency between 2 circles and if I give you a choice of showing this tangency, you will always/instinctly choose the horizontal scenario as it is more straightforward to visualize.

Hope this helps.

Sure thanks. The reason, I like to show one other example besides horizontal is to show that this isn't something special which occurred because the lines were 100% horizontal it's something true in 100% of scenarios. So I usually like to show a couple of examples. One the obvious one and another the less obvious way.
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Re: Two Tangent circles A and B have radii r1 and r2, respectively. What  [#permalink]

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