What is the easiest way to do this? | Mathematics | Page 1

spiral bison Jan 1, 2025, 3:54 AM

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cyan cargoBOT Jan 1, 2025, 3:54 AM

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spiral bison Jan 1, 2025, 3:56 AM

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this looks easy but i actually dont really know what to do

oblique radish Jan 1, 2025, 4:04 AM

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spiral bison

The distance between the points A and B is sqrt(8)

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Form two simultaneous equations using this info with the variables being the x and y coordinates of C

spiral bison Jan 1, 2025, 4:06 AM

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so solve (x-2)^2 +y^2 = 8; x^2+(y-2)^2 = 8?

stoic kite Jan 1, 2025, 4:06 AM

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so that's the general method for finding the corners of an equilateral triangle

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but here you can do better

oblique radish Jan 1, 2025, 4:06 AM

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Yea but maybe there's an easier way

spiral bison Jan 1, 2025, 4:07 AM

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what would be the trick to do the smarter way?

stoic kite Jan 1, 2025, 4:07 AM

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notice that an equilateral triangle has reflectional symmetry about all its 3 axis

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more importantly, what that means is that if we know 2 points of an equilateral triangle then the 3rd point must lie on the midsegment between the two points

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(this is also true for isosceles triangles, if you choose the starting 2 points to be the base points)

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so, what is the midsegment of A and B?

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you shouldn't need to do any math to tell me this

spiral bison Jan 1, 2025, 4:14 AM

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(1,1)?

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how do you get the 2 points from this symmetry tho?

stoic kite Jan 1, 2025, 4:15 AM

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the midsegment is a line

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not a point

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it's the line between A and B

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so notice that if I flip the plane around the line x=y , A is sent to B and B is sent to B

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therefore, whereever the line between A and B is, it must remain the same before and after you flip the plane around the line x = y

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so what can it be

spiral bison Jan 1, 2025, 4:17 AM

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are you asking for the equation of the line AB?

stoic kite Jan 1, 2025, 4:18 AM

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no, I'm asking for the midsegment

stoic kite Jan 1, 2025, 4:18 AM

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stoic kite it's the line between A and B

okay this sounds wrong

stoic kite Jan 1, 2025, 4:19 AM

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spiral bison

but it's the green line here

spiral bison Jan 1, 2025, 4:19 AM

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so the equation of the green line?

stoic kite Jan 1, 2025, 4:19 AM

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yes

spiral bison Jan 1, 2025, 4:19 AM

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is it just y =x?

stoic kite Jan 1, 2025, 4:20 AM

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yes

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so what does that tell you?

spiral bison Jan 1, 2025, 4:21 AM

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C must be on that line

stoic kite Jan 1, 2025, 4:21 AM

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yes, but we already know that

spiral bison Jan 1, 2025, 4:22 AM

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B is reflection of A?

stoic kite Jan 1, 2025, 4:22 AM

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what can you tell me about the coordinates of C

spiral bison Jan 1, 2025, 4:23 AM

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i dont really know

stoic kite Jan 1, 2025, 4:23 AM

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C must be on x=y

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so let's say the x coordinate of C is 1. what is its y coordinate

spiral bison Jan 1, 2025, 4:24 AM

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1?

stoic kite Jan 1, 2025, 4:24 AM

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exactly

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so the coordinates of C is exactly (a,a) for some a

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and so we only need to find one coordinate of C, and we get the other for free

spiral bison Jan 1, 2025, 4:27 AM

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ohh i think i see it now

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do i just do this and apply pythagorean theorem?

stoic kite Jan 1, 2025, 4:28 AM

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yeah you can do that

spiral bison Jan 1, 2025, 4:28 AM

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to find the distance from (2,0)

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is there another way?

stoic kite Jan 1, 2025, 4:28 AM

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that's good enough

spiral bison Jan 1, 2025, 4:29 AM

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was there some super secret method lol?

stoic kite Jan 1, 2025, 4:29 AM

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no not really for this part

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C is at (a,a)

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distance between A and B is 2 sqrt2

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so distance between A and C is 2sqrt2

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solve for that

spiral bison Jan 1, 2025, 4:31 AM

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ok thanks so much then, so for these type of problems with equilateral or isosceles, you should find the line of midsegment first to get the relationship between x and y?

stoic kite Jan 1, 2025, 4:31 AM

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depends

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in this problem you should do that, because you don't need to do any math to be able to find the midsegment immediately

spiral bison Jan 1, 2025, 4:32 AM

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btw how do you find the midsegment without doing math lol, i kinda just did negative reciprocal of the slope of line AB and find the intercept

stoic kite Jan 1, 2025, 4:33 AM

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stoic kite so notice that if I flip the plane around the line x=y , A is sent to B and B is...

it's this

spiral bison Jan 1, 2025, 4:33 AM

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i see, so this is where the symmetry comes in play

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thx so much my teacher didnt really teach us this yet.

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.close

cyan cargoBOT Jan 1, 2025, 4:39 AM

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Solved

Post marked as solved by @spiral bison.

Use .unsolved if this was a mistake.

spiral bison Jan 1, 2025, 4:41 AM

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actually i can just use the 30-60-90 triangle instead of pythagorean theorem

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@stoic kite I keep getting a = 2 (+-) sqrt(2), but i tried the general way and got 1 (+-) sqrt(3).

stoic kite Jan 1, 2025, 4:54 AM

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spiral bison <@530674466028650516> I keep getting a = 2 (+-) sqrt(2), but i tried the general...

can you show your working

spiral bison Jan 1, 2025, 4:54 AM

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i use the 30-60-90 triangle and i get the distance as sqrt(2)

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and then the offset is 2

stoic kite Jan 1, 2025, 4:55 AM

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what?

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this doesn't make any sense

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can you explain a little more

spiral bison Jan 1, 2025, 4:56 AM

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give me a second i'll draw it out

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the green triangle is 30-60-90

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and the sidelength is 2sqrt(2), the red part is sqrt(2)

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@stoic kite did i do something wrong?

stoic kite Jan 1, 2025, 5:00 AM

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I don't want pictures can you show me your math

spiral bison Jan 1, 2025, 5:01 AM

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ok AB = 2sqrt(2)

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and then AC is also 2sqrt(2).

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The red part is 1/2 * 2sqrt(2) = sqrt(2) with special right triangles

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the distance from the x=0 is 2 + sqrt(2)

stoic kite Jan 1, 2025, 5:05 AM

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why is red part 1/2 * 2sqrt(2)

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it's not

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whenever you want to find the distance formula between 2 points, use the distance formula

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don't do anything fancy with Pythagorean triangles

spiral bison Jan 1, 2025, 5:07 AM

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i see mb i thought the green triangle was 30-60-90, my picture was very wrong lmao

stoic kite Jan 1, 2025, 5:07 AM

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do not trust pictures

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never trust pictures. only math

#What is the easiest way to do this?