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?
To find the center of the circle circumscribed around triangle ABC, we can use the properties of perpendicular bisectors.
Step 1: Find the midpoints of the sides of the triangle.
The midpoint of a line segment can be found by taking the average of the x-coordinates and the average of the y-coordinates.
For side AB:
Midpoint of AB = ( (x1 + x2) / 2, (y1 + y2) / 2)
Midpoint of AB = ( (-8 + 0) / 2, (0 + 0) / 2)
Midpoint of AB = (-4, 0)
For side BC:
Midpoint of BC = ( (x1 + x2) / 2, (y1 + y2) / 2)
Midpoint of BC = ( (0 + 0) / 2, (0 + (-4)) / 2)
Midpoint of BC = (0, -2)
For side AC:
Midpoint of AC = ( (x1 + x2) / 2, (y1 + y2) / 2)
Midpoint of AC = ( (-8 + 0) / 2, (0 + (-4)) / 2)
Midpoint of AC = (-4, -2)
Step 2: Find the slopes of the sides of the triangle.
The slope of a line can be found using the formula (y2 - y1) / (x2 - x1).
Slope of AB:
Slope of AB = (0 - 0) / (0 - (-8)) = 0 / 8 = 0
Slope of BC:
Slope of BC = (-4 - 0) / (0 - 0) = -4 / 0 (undefined)
Slope of AC:
Slope of AC = (-4 - 0) / (0 - (-8)) = -4 / 8 = -1/2
Step 3: Find the slopes of the perpendicular bisectors.
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
Perpendicular bisector of AB:
Perpendicular slope of AB = -1 / slope of AB = -1 / 0 (undefined)
Perpendicular bisector of BC:
Perpendicular slope of BC = -1 / slope of BC = -1 / (-4 / 0) (undefined)
Perpendicular bisector of AC:
Perpendicular slope of AC = -1 / slope of AC = -1 / (-1/2) = 2
Step 4: Find the equations of the perpendicular bisectors.
The equation of a line can be found using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is its slope.
Equation of perpendicular bisector of AB passing through (-4, 0):
y - 0 = undefined(x - (-4))
x = -4
Equation of perpendicular bisector of BC passing through (0, -2):
y - (-2) = undefined(x - 0)
y = -2
Equation of perpendicular bisector of AC passing through (-4, -2):
y - (-2) = 2(x - (-4))
y + 2 = 2(x + 4)
y = 2x + 10
Step 5: Find the point of intersection of two perpendicular bisectors.
The center of the circumscribed circle will be the point of intersection of the perpendicular bisectors.
To find the intersection points, we can equate the equations of two perpendicular bisectors.
x = -4 (from Equation of perpendicular bisector of AB)
y = -2 (from Equation of perpendicular bisector of BC)
Therefore, the center of the circle circumscribed around triangle ABC is located at (-4, -2).