1the new vertices are (-1,1) (2,1-√3) (2,1+√3)is this correct
A triangle with vertices at 3+i, i(1+√3) , and i(1-√3) is rotated about its circumcentre through an angle of Î /3 in anticlockwise direction.Find the new positon of the vertices.
Pls post the method and method to find circumcentre
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9 Answers
39Coords of vertices of triangle are A(3, 1), B(0, 1 + √3) and C(0, 1 - √3).
Lengths are a = AB = √9 + 3 = √12
b = BC = √12
c = AC = √12
Thus this is an equilateral triangle.
The circumcentre coincides with the centroid luckily here. Otherwise it is a point which is equidistant from all sides of the triangle.
So our circumcentre is (1,1).
I'm not good in rotation of axes...but I'll try. Since we are rotating the axes wrt circumcentre, it is our origin. Our shift of origin is from (1,1) to (0,0). (To make rotation easier)
0 = X, 0 = Y.
1 = x, 1 = y.
X = x+h, Y = y+k
Thus our new circumcentre is (-1, -1).
Similarly the vertices become (2, 0) , (-1, √3) and (-1, -√3).
Angle of rotation = +60°(in ACW direction).
Let us take point A to rotate.
The old point is (x,y) = (2,0) [As per our new origin]
x = Xcos60 - Ysin60
y = Xsin60 + Ycos60
where (X, Y) are the new coords.
2 = X/2 - Y√3/2
0 = X√3/2 + Y/2
Multiply by 2,
4 = X - √3Y
0 = √3X + Y
or
4√3 = X√3 - 3Y
0 = 3√3X + 3Y
Adding,
4√3 = 4√3X or X = 1, Y = -√3
So A(1, -√3). Add 1,1 to shift back to old origin to get (2, 1 - √3).
Similarly do the rest? lola. There might be a shorter method but...this is all that occurred to me so sorry.
21array use rotation theorem
A(3+i), B i(1 + √3) and C i(1 - √3).
1+i is ur circumcenter and points A B C lie on a circle with center z0=1+i
new rotated positions wrt to center will also lie on same circle
now u hav to rotate A B C wrt to center
let ZAbe new rotated vertex of A
so from coni theorem
ZA-(1+i)(3+i)-(1+i )=eiπ/3
similarly do for oders.
21@pritish
its not something extraordinary...
u can see it in any coaching material.......btw der is a whole exercise on this theorem in MLk..
aur agar isko nai pada toh samajhlo complex nahi pada. ;)
i cud find this link on net see it - http://bit.ly/ae9HpJ
39lola nahi padha tha tabhi toh poocha maine re! thanks for the link.