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The vertices of a triangle are the points of intersection of the line y=-x-1, the line x=2, and the line y=2. Find the center of the circle passing through all three vertices. Enter the coordinates as an ordered pair.

 Apr 19, 2024
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Find the intersection points:

 

The line x = 2 intersects the y-axis at (2, 0).

 

To find the intersection of y = -x - 1 and y = 2, we set the y values equal: -x - 1 = 2 --> x = -3. Substituting this x value back into y = -x - 1 gives the point (-3, -2).

 

Find the midpoint of each side:

 

Midpoint of side connecting (2, 0) and (-3, -2):

 

x-coordinate: ((2) + (-3)) / 2 = -1/2

 

y-coordinate: ((0) + (-2)) / 2 = -1

 

Midpoint is (-1/2, -1).

 

Since the line x = 2 is a vertical line, any point on this line with a y-coordinate of -1 will be the midpoint of the side connecting (2, 0) and the intersection point on this line (which doesn't matter since it's a vertical line). Therefore, the midpoint is (2, -1).

 

Since all three vertices lie on the line x = 2, the center of the circle that passes through all three vertices must also lie on this line. Therefore, the center of the circle has an x-coordinate of 2.

 

Since the midpoint of one side is also the midpoint of another side that lies on the same line, the center of the circle must also coincide with the midpoint we found: (2, -1).

 

Therefore, the center of the circle is at (2,−1)​.

 Apr 19, 2024

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