Please help me solve this and usually whats the method to provide proofs in questions regarding geometry like in other places u can use induction, contradiction and stuff ?
#Geometry question
lethal verge
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Please help me solve this and usually whats the method to provide proofs in ques...
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silent wren
Please help me solve this and usually whats the method to provide proofs in ques...
Well, take the points as they are given: K (0, 0), D (d, 0), A (a1, a2), W (b1, b2). Then see the coordinates of everything else.
lethal verge
Well, take the points as they are given: K (0, 0), D (d, 0), A (a1, a2), W (b1, ...
Im doing some stuff but idk where its leading to
silent wren
Hm...
Yeah, maybe this isn't the best approach.
lethal verge
Yes i found i did find midpoint co-ordinates here but the only other information provided in the question is abt distance
And that doesn’t feel right
silent wren
Hm. Maybe it could be easier in terms of vectors?
lethal verge
Cross product ?
silent wren
As in, suppose vector AW = u, vector DK = v, vector MN = w.
I believe you should be able to express w in terms of u and v. And we must have 2|w| = |u| + |v|, so maybe we can get something in terms of that.
Oh yeah, this will work, try it.
lethal verge
Ohk i'll give it a try
quick kestrel
Wich level??
silent wren
Level? What do you mean?
lethal verge
Wich level??
Grade 12 ig ?
silent wren
As in, suppose vector AW = u, vector DK = v, vector MN = w.
I believe you should...
Any success with this approach?
lethal verge
Im yet to find magnitudes
lethal verge
silent wren
You don't need to find any magnitudes.
lethal verge
You don't need to find any magnitudes.
Oh
lethal verge
So this is very over-complicated right ?
quick kestrel
Grade 12 ig ?
Oh i m grade 9 😦
quick kestrel
U can give it a try if u want
okk i ll try..
silent wren
So this is very over-complicated right ?
Yeah.
lethal verge
Ok i understood
Just gimme a min
Lemme send
I did this
Found u, v and w
Then used the info 2w = u + v
And both sides equated
silent wren
Sure, that works. Though, here's an even simpler approach.
We have (everything is a vector here):
u = AW
v = DK
w = MN = OM - ON = (1/2)(OK + OW) - (1/2)(OA + OD) = (1/2)(OW - OA) + (1/2)(OK - OD) = (1/2)(AW + KD) = (1/2)(u - v)
We must have |w| = (1/2)(|u| + |v|). So:
|(1/2)(u - v)| = (1/2)(|u| + |v|)
|u - v| = |u| + |v|
Let's square both sides.
u^2 + v^2 - 2u·v = u^2 + v^2 + 2|u||v|
u·v = -|u||v|
But this just means u and v are antiparallel, hence lines AW and DK are parallel.
silent wren
Found u, v and w
Yeah, exactly. Though, as you can see, there's no need for coordinates.
As in, the initial points can be absolutely anywhere in the plane.
lethal verge
U used dot product ?
silent wren
Yeah.
silent wren
Where did u get 180° from ?
Recall that cos(θ) = u·v/(|u||v|).
So, if u·v = -|u||v|, what will cos(θ) be?
lethal verge
-1
silent wren
Yup.
lethal verge
And θ needs to be 180° for that
silent wren
Yeah.
silent wren
What do you mean?
lethal verge
That shows theyre parallel ?
silent wren
Obviously.
lethal verge
Ye it makes sense
silent wren
If the angle between vectors is 180°, then they have opposite directions, so they must lie on parallel lines.
lethal verge
I used cross product to show parallel
silent wren
Well, maybe that's also possible here.
lethal verge
If the angle between vectors is 180°, then they have opposite directions, so the...
This can be used in lines as well right ?
L = a + λb
silent wren
This can be used in lines as well right ?
Well, the angle between lines is usually taken between 0 and π/2, rather.
For vectors it's between 0 and π.
lethal verge
Oh i see
lethal verge
Well, the angle between lines is usually taken between 0 and π/2, rather.
My point was we can use dot product everywhere right ?
Cuz its much simpler to calculate as compared to cross product
For lines if |u||v| = u•v
Theyre parallel
?
silent wren
My point was we can use dot product everywhere right ?
I mean, it depends on the problem.
Sometimes dot product is easier, sometimes cross product is easier, and sometimes you need other approaches.
lethal verge
Dam ok
lethal verge
Sure, that works. Though, here's an even simpler approach.
We have (everything i...
I'll have a look at this solution
lethal verge
But this also works right ?
I'll obv go for the better method
But just wanna know if the solution is atleast correct
silent wren
But this also works right ?
Yeah, this seem to be pretty much the same, just with explicit coordinates.
lethal verge
Ok
lethal verge
Yeah, this seem to be pretty much the same, just with explicit coordinates.
Thanks
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silent wren
You're welcome!
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