Question 53767: Verify that these three points: (0,0) and (2a,0) and (a, a*sqrt3) are the vertices of an equilateral triangle. Then, when you get that done, show that the midpoints of the three sides are the vertices of a second equilateral triangle.
I did the first part of the problem, but I can't get my numbers right to show that the midpoints are equilateral. Please help!
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Verify that these three points: (0,0)=A SAY and (2a,0)=B SAY and (a, a*sqrt3)=C SAY are the vertices of an equilateral triangle.
AB = SQRT(4A^2)=2A
BC= SQRT(A^2+3A^2)=2A
CA=SQRT(A^2+3A^2)=2A
HENCE ABC IS EQUILATERAL TRIANGLE
Then, when you get that done, show that the midpoints of the three sides are the vertices of a second equilateral triangle.
LET THE MIDPOINTS OF AB,BC,CA BE D,E,F RESPECTIVELY.
D = [(0+2A)/2,(0+0)/2]=(A,0)
E= [(2A+A)/2,(0+ASQRT(3))/2]=[3A/2,ASQRT(3)/2]
F=[A/2,ASQRT(3)/2]
DE^2 = (3A/2 - A)^2 + (ASQRT(3)/2)^2=A^2/4 +3A^2/4 = A^2
EF^2 = [(3A/2 -A/2)^2 + 0^2]=A^2
FD^2 = [(A-A/2)^2 + {ASQRT(3)/2}^2] = A^2/4 +3A^2/4= A^2
HENCE DE=EF=FD=A..
DEF IS AN EQUILATERAL TRIANGLE
I did the first part of the problem, but I can't get my numbers right to show that the midpoints are equilateral. Please help!
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