dan815
  • dan815
Prove height of isosceles intersects the midpoint of the base @owlcoffee
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
dan815
  • dan815
|dw:1435253588480:dw|
Owlcoffee
  • Owlcoffee
So, here I go: First of I'll list up the hypothesis, which is the given information: (H) ABC isoceles. (T) AX = BX |dw:1435253753194:dw| To begin, I'll take the very definition of isoceles triangle: \[(1) CA=CB\] and : \[(2)
alekos
  • alekos
the two base angles, say α, are equal so tanα = h/b1 = h/b2 where b1 + b2 = base therefore b1 = b2

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dan815
  • dan815
show they are equal too for isoc
dan815
  • dan815
: )
dan815
  • dan815
ill allow the usage of AAS SSS and SAS only
alekos
  • alekos
the ancient greeks proved this 2000 years ago :)
dan815
  • dan815
yup
dan815
  • dan815
using tan -.- cheater face
dan815
  • dan815
thats a high end function
dan815
  • dan815
|dw:1435255353420:dw|
dan815
  • dan815
and so show the base angles of isosceles|dw:1435255424617:dw| are equal we can do
dan815
  • dan815
still kind of cheating because im using SSS , but it would be really nice to see without that, or show that proof too
Owlcoffee
  • Owlcoffee
I really like your method Dan, Ill have that in my repertoire.
dan815
  • dan815
if u like these kind of examples i have a fun one for u
dan815
  • dan815
you can try to solve this problem with only geometry its a famous one dont look it up xD
Owlcoffee
  • Owlcoffee
Give me your best shot.
dan815
  • dan815
|dw:1435255751221:dw|
dan815
  • dan815
|dw:1435255859970:dw|
dan815
  • dan815
|dw:1435255900735:dw|
dan815
  • dan815
here u go :)
dan815
  • dan815
think of the line as water, the distance from the city A to the water is a and City B to water is b and the horizontal distance from city A to City B is L
dan815
  • dan815
you must draw water from the river bank and go to the next city in the shortest route possible
dan815
  • dan815
is the question clear?
Owlcoffee
  • Owlcoffee
Yes
dan815
  • dan815
okay let a = 5, b= 10, L = 20 find point C|dw:1435256121850:dw|
Owlcoffee
  • Owlcoffee
|dw:1435256155530:dw| Let a ortonormated reference system exist, such way that the segment "a" intersection with the line "L" defines the origin. Therefore: \[A(0,5)\] \[B(20, 10)\] \[C(x_c , y_c)\] But since C belongs in my actual x_axis, then: \[C(x_c, 0)\] So if I find the line from C to A, then: \[m=\frac{ 0-x_c }{ 5-0 }=\frac{ -x_c }{ 5 }\] \[t)(y-5)=(\frac{ -x_c }{ 5 })(x)\] \[t)xx_c+2y-25=0\] And If I find the other line which is composed by C and B: \[m=\frac{ 20-x_c }{ 10-0 }\] \[r)(y-10)=(\frac{ 20-x_c }{ 10 })(x-20)\] \[r)10y-100=20x-400-xx_c-20x_c\] \[r)(x_c-20)x+10y+(20x_c+300)=0\] So, the intersection of t) , r) and the line y=0 must give me as a result the x-coordinate of the point C in my reference system.
dan815
  • dan815
|dw:1435260599276:dw|
dan815
  • dan815
what you do is reflect a across the horizontal line, and connect point b to reflection of a, now this is a straight line and you will see that any other point along this river will yield a line bigger
dan815
  • dan815
|dw:1435260688944:dw|
dan815
  • dan815
isnt it pretty :)
Owlcoffee
  • Owlcoffee
Wow, much simpler than I thought. And you used euclidean geometry... Reflections...
dan815
  • dan815
it's one of the most elegant solutions from history
Owlcoffee
  • Owlcoffee
With no wonder. Very straight and elegantly solved.
dan815
  • dan815
haha its looks very simple once u see it, but boy ill tell you, its very rare someone would find this solution
dan815
  • dan815
especially nowadays when everyone just resorts to calculus
dan815
  • dan815
this one can be solved as a minimization problem, and gets all messy
dan815
  • dan815
and an interesting thing comes which is that these angles are equal
dan815
  • dan815
|dw:1435261128090:dw|
dan815
  • dan815
so the point always such that those angles are equal, so these 2 triangles are similiar
dan815
  • dan815
and the main triangle here can be draw too to solve completely
Owlcoffee
  • Owlcoffee
Well yes, they are equal because of vertical angles, and since one is the reflection, the third must be equal as well.
dan815
  • dan815
|dw:1435261208966:dw|
dan815
  • dan815
|dw:1435261234039:dw|
Owlcoffee
  • Owlcoffee
proportionality. I really need to dig back to geometry, I am starting to get rusty.
dan815
  • dan815
theres multiple ways to solve it but this is the mains solution |dw:1435261434618:dw|
dan815
  • dan815
once u draw this pic u are pretty much done technically, as u just managed to draw shortest line
dan815
  • dan815
in books, they usually stop there
Owlcoffee
  • Owlcoffee
Yes, indeed. I think when I used analytical geometry i limited myself to finding the point C. And I can understand why people resort to calculus, you can translate it to a function and find the minima.
dan815
  • dan815
there an extension to this, which is in higher dimension using same idea
dan815
  • dan815
or like if u have to hit 2 lines
Owlcoffee
  • Owlcoffee
In 3 dimensions?
dan815
  • dan815
|dw:1435261603447:dw|
dan815
  • dan815
like this one, very similar , but now u have to hit 2 lines before going from town a to b
dan815
  • dan815
maybe u can find this one :) and get back to me tomorrow
Owlcoffee
  • Owlcoffee
I will try, I believe you have many questions to answer nowdays :x

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