#Help with geometry

63 messages · Page 1 of 1 (latest)

gusty ivy
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Hello people, I’m facing yet another math problem. So here is the issue - we’re writing a test tomorrow and I couldn’t come to math class cause I was sick. Now, I’m trying to review the material before the test and I simply don’t even know how one should approach the problem I’m facing. Here is the problem:

You are given a circle k, with center S and radius r. Then you are also given a circle m, with center O and radius v. Find all line segments XY, such that X belongs to k and Y belongs to m, XY is parallel to SO and also |XY| = 1/2|SO|, where |x| represents the length of the segment.

Any hints would be greatly appreciated

drowsy tundraBOT
gusty ivy
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<@&286206848099549185>

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!15m

drowsy tundraBOT
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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

brazen marsh
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you have to do this for specific values? The general solution is going to be pretty complicated

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Start by listing the equations implied by the constraints

gusty ivy
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The sad part is that my teacher just gave us like 10+ excercises like this with no additional information given and expects us to pass the exam

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the last proper topic we kind of "covered" were transformations and symmetry

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here is an image I constructed

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with values that I chose

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I'm guessing its just a bunch of casework with some neat trick

brazen marsh
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you usually working in coordinates? or vectors?

gusty ivy
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we usually do constructions with like ruler and a compass

brazen marsh
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without loss of generality, you could assume the circles are oriented this way and assume x coordinate of each circle is 0

X= (x, y(S)+-sqrt(k^2-x^2))
Y = (x, y(O)+-sqrt(v^2-x^2))

distance between becomes |SO| +-sqrt(k^2-x^2) +-sqrt(v^2-x^2) = |SO|/2
solvable

gusty ivy
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yea the other person that responded gave me pretty much similiar solution

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but wait

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let me think about this

brazen marsh
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there might be some geometric construction but that's not how I usually attack problems

gusty ivy
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i don't think its entirely correct

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the X, Y don't usually have to be on the same x coordinate

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but I see the point

brazen marsh
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they do if you hold the assumptions I made:
without loss of generality, you could assume the circles are oriented this way and assume x coordinate of each circle is 0

gusty ivy
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okay, so you essentially "translate" the image to the origin

brazen marsh
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for generic circles, you'd end up doing a rotation and transposition

gusty ivy
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but what I'm not getting is that the solution could involve

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ok assume X, Y are in the same places as S and O

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what could essentially happen is that you move both by a vector, that is diagonal

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and that yields the correct solution

brazen marsh
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X and Y in the same place as S and O would mean 0 radius

gusty ivy
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move = translate

brazen marsh
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are you doing algebra like this regularly in class, I'm starting to suspect this is a compass and ruler problem

gusty ivy
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we are currently doing "construction type problems"

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but I know something about vectors, dot product, cross product and euclidean geometry

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because I have strong competetive programming background

gusty ivy
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but I couldn't even come up with an analytical solution

brazen marsh
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i don't this is sufficient but you can construct parallel lines to the line from S to O through points on one circle and measure them and then make small adjustments until you have a length of |SO|/2

gusty ivy
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ok but thats kind of stupid

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thats not really a solution

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cause like thats not precise

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thats approximated

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like if that was possible i'd just binary search the solution

brazen marsh
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what can I say I'm a numerical guy. My frankensteined super computer is named Taylor

gusty ivy
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taylor no swift

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taylor series

tropic shard
brazen marsh
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an elegant solution

tropic shard
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you couldve just considered every point that is half of SO above m, thats where you can get the solution from

brazen marsh
gusty ivy
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Oh wait they don’t

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wow thats such a nice solution

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tysm for your help!!

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❤️

tropic shard
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.solved
np