Question
Find the circumcenter of ΔABC with A(1, 6), B(1, 4), and C(5, 4).
(1 point)
Responses
(5, 3)
(5, 3)
(3, 5)
(3, 5)
(7, 3)
(7, 3)
(1, 7)
(1, 7)
(1 point)
Responses
(5, 3)
(5, 3)
(3, 5)
(3, 5)
(7, 3)
(7, 3)
(1, 7)
(1, 7)
Answers
Answer
no it is not its (3,5)
Answered by
GPT 3.5
To find the circumcenter of a triangle, we first need to find the midpoint of each side of the triangle, then find the perpendicular bisector of each side. The circumcenter is the point where all three perpendicular bisectors intersect.
Let's begin by finding the midpoints of the sides AB, BC, and AC.
Midpoint of AB:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of AB = ((1 + 1)/2, (6 + 4)/2) = (1, 5)
Midpoint of BC:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of BC = ((1 + 5)/2, (4 + 4)/2) = (3, 4)
Midpoint of AC:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of AC = ((1 + 5)/2, (6 + 4)/2) = (3, 5)
Next, let's find the slopes of the lines AB, BC, and AC.
Slope of AB:
Slope = (y2 - y1)/(x2 - x1)
Slope of AB = (4 - 6)/(1 - 1) = undefined
Slope of BC:
Slope = (y2 - y1)/(x2 - x1)
Slope of BC = (4 - 4)/(5 - 1) = 0
Slope of AC:
Slope = (y2 - y1)/(x2 - x1)
Slope of AC = (4 - 6)/(5 - 1) = -1
Since the slope of AB is undefined, the perpendicular bisector of AB is a vertical line passing through the midpoint (1, 5).
Since the slope of BC is 0, the perpendicular bisector of BC is a horizontal line passing through the midpoint (3, 4).
Since the slope of AC is -1, the perpendicular bisector of AC also has a slope of -1. We can find its equation by using the midpoint (3, 5).
Equation of the perpendicular bisector of AC:
y - y1 = m(x - x1)
y - 5 = -1(x - 3)
y - 5 = -x + 3
y = -x + 8
Now we have the equations of the perpendicular bisectors of AB, BC, and AC. To find the circumcenter, we need to find the point where these lines intersect.
Solving the system of equations formed by the perpendicular bisectors, we get the point (5, 3) as the circumcenter of triangle ABC.
Thus, the correct response is:
(5, 3)
Let's begin by finding the midpoints of the sides AB, BC, and AC.
Midpoint of AB:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of AB = ((1 + 1)/2, (6 + 4)/2) = (1, 5)
Midpoint of BC:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of BC = ((1 + 5)/2, (4 + 4)/2) = (3, 4)
Midpoint of AC:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint of AC = ((1 + 5)/2, (6 + 4)/2) = (3, 5)
Next, let's find the slopes of the lines AB, BC, and AC.
Slope of AB:
Slope = (y2 - y1)/(x2 - x1)
Slope of AB = (4 - 6)/(1 - 1) = undefined
Slope of BC:
Slope = (y2 - y1)/(x2 - x1)
Slope of BC = (4 - 4)/(5 - 1) = 0
Slope of AC:
Slope = (y2 - y1)/(x2 - x1)
Slope of AC = (4 - 6)/(5 - 1) = -1
Since the slope of AB is undefined, the perpendicular bisector of AB is a vertical line passing through the midpoint (1, 5).
Since the slope of BC is 0, the perpendicular bisector of BC is a horizontal line passing through the midpoint (3, 4).
Since the slope of AC is -1, the perpendicular bisector of AC also has a slope of -1. We can find its equation by using the midpoint (3, 5).
Equation of the perpendicular bisector of AC:
y - y1 = m(x - x1)
y - 5 = -1(x - 3)
y - 5 = -x + 3
y = -x + 8
Now we have the equations of the perpendicular bisectors of AB, BC, and AC. To find the circumcenter, we need to find the point where these lines intersect.
Solving the system of equations formed by the perpendicular bisectors, we get the point (5, 3) as the circumcenter of triangle ABC.
Thus, the correct response is:
(5, 3)
Answered by
GPT 3.5
Apologies for the mistake in my previous response. You are correct, the correct circumcenter of triangle ABC with vertices A(1, 6), B(1, 4), and C(5, 4) is indeed (3, 5). Thank you for bringing that to my attention and I apologize for any confusion caused.