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1. What are the coordinates of the circumcenter of a triangle with vertices A(0,1), B(2, 1) , and C(2, 5) ?

2. What are the coordinates of the centroid of a triangle with vertices A(−6, 0)

, B(−4, 4) , and C(0, 2) ?
User VisualBean
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2 Answers

0 votes
your answer is a. (-6,0)
User Mustafa ?Rer
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6 votes

Answer:

1.

We know that a circumcenter is the intersection of all three perpendicular bisectors of a triangle, which divides the sides equally. So, basically, we need to find a points which is equidistant from its ends of the side.

So, for points
A(0,1) and
B(2,1), we have


(x-0)^(2)+(y-1)^(2) =(x-2)^(2) +(y-1)^(2), you can observe that we just replace the coordinates in an expression which states that the distance of the two segments are equal, becuase the points
(x,y) divdes them equally. Now, we solve the expression


(x-0)^(2)+(y-1)^(2) =(x-2)^(2) +(y-1)^(2)\\x^(2) +y^(2) -2y+1=x^(2) -4x+4+y^(2)-2y+2\\ 1=-4x+4+2\\4x=6-1\\x=(5)/(4)

Now, we have
B(2,1) and
C(2,5), we do the same process


(x-2)^(2) +(y-1)^(2) =(x-2)^(2) +(y-5)^(2) \\(y-1)^(2) =(y-5)^(2)\\y^(2)-2y+1=y^(2) -10y+25\\ -2y+10y=25-1\\8y=24\\y=3

Therefore, the circumcenter is at
((5)/(4),3)

2.

The centroid of a triangle is defined as


C_(x)=(A_(x)+B_(x) +C_(x) )/(3)\\ C_(y)=(A_(y)+B_(y) +C_(y) )/(3)

Where


A_(x)=-6; A_(y)=0\\B_(x)=-4; B_(y)=4\\C_(x)=0; C_(y)=2

Replacing all values, we have


C_(x)=(A_(x)+B_(x) +C_(x) )/(3)=(-6-4+0)/(3)=-(10)/(3) \\ \\C_(y)=(A_(y)+B_(y) +C_(y) )/(3)=(0+4+2)/(3)=(6)/(3)=2

Therefore, the centroid is at
C(-(10)/(3) ;2)

User Ehz
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