The vertices of ABC are A(-8, 0), B(0, 0), and C(0, -4). If a circle is circumscribed around ABC, where would the center of the circle be located?
All I can suggest to do is draw it. I got (-3,-2)..im sure it's something like that.
To find the center of the circle circumscribed around triangle ABC, we can make use of the fact that the center lies at the intersection of the perpendicular bisectors of the triangle's sides.
Step 1: Find the midpoints of the triangle's sides.
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formulas:
Midpoint_x = (x1 + x2) / 2
Midpoint_y = (y1 + y2) / 2
For side AB with endpoints A(-8, 0) and B(0, 0):
Midpoint_AB_x = (-8 + 0) / 2 = -4
Midpoint_AB_y = (0 + 0) / 2 = 0
For side BC with endpoints B(0, 0) and C(0, -4):
Midpoint_BC_x = (0 + 0) / 2 = 0
Midpoint_BC_y = (0 + (-4)) / 2 = -2
For side AC with endpoints A(-8, 0) and C(0, -4):
Midpoint_AC_x = (-8 + 0) / 2 = -4
Midpoint_AC_y = (0 + (-4)) / 2 = -2
Step 2: Find the slopes of the triangle's sides.
The slope of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:
Slope = (y2 - y1) / (x2 - x1)
For side AB:
Slope_AB = (0 - 0) / (0 - (-8)) = 0 / 8 = 0
For side BC:
Slope_BC = (-4 - 0) / (0 - 0) = (-4) / 0 (undefined)
For side AC:
Slope_AC = (-4 - 0) / (0 - (-8)) = (-4) / 8 = -0.5
Step 3: Find the equations of the perpendicular bisectors.
The equation of a line perpendicular to a line with slope m and passing through a point (x0, y0) is given by the formula:
Perpendicular Bisector: y - y0 = (-1/m)(x - x0)
For side AB:
Since the slope is 0, the perpendicular bisector is a vertical line passing through the midpoint AB:
Equation_AB: x = -4
For side BC:
Since the slope is undefined, the perpendicular bisector is a horizontal line passing through the midpoint BC:
Equation_BC: y = -2
For side AC:
The perpendicular bisector of AC will have a slope that is the negative reciprocal of the slope of AC (which is -0.5):
m_perpendicular = -1 / (-0.5) = 2
Using the midpoint AC (-4, -2) and the slope 2, we can find the equation of the perpendicular bisector:
Equation_AC: y - (-2) = 2(x - (-4))
Simplifying: y + 2 = 2x + 8
Rearranging: 2x - y = -6
Step 4: Find the intersection point of the perpendicular bisectors.
To find the center of the circle, we need to find the point where the perpendicular bisectors intersect. In this case, since Equation_AB is vertical and Equation_BC is horizontal, the intersection point will be the point (-4, -2) that lies on both lines.
Therefore, the center of the circle circumscribed around triangle ABC is located at (-4, -2).