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)
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)
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.
To find the circumcenter of triangle ABC, we can follow these steps:
1. Find the midpoints of two sides of the triangle.
The midpoint of side AB can be found by averaging the x-coordinates and the y-coordinates separately:
Midpoint AB = ((1+1)/2, (6+4)/2) = (1, 5)
The midpoint of side BC can be found the same way:
Midpoint BC = ((1+5)/2, (4+4)/2) = (3, 4)
2. Find the slopes of the perpendicular bisectors of these two sides.
The slope of the perpendicular bisector of side AB is the negative reciprocal of the slope of AB. Let's find the slope of AB first:
Slope AB = (5-6)/(1-1) = undefined
So, the slope of the perpendicular bisector is 0.
The slope of the perpendicular bisector of side BC can be found the same way:
Slope BC = (4-4)/(3-1) = 0
3. Write the equations of these perpendicular bisectors.
The equation of the perpendicular bisector of AB is of the form x = 1.
The equation of the perpendicular bisector of BC is of the form y = 4.
4. Find the intersection point of these two perpendicular bisectors, which is the circumcenter of triangle ABC.
Since x = 1 and y = 4 satisfy both equations, the intersection point is (1, 4).
Therefore, the circumcenter of triangle ABC is (1, 4).
The correct response is (1, 4).
To find the circumcenter of a triangle, you can use the following steps:
1. Find the midpoints of two sides of the triangle.
- In this case, find the midpoint of side AB and side AC.
Midpoint_AB = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint_AC = ((x1 + x3)/2, (y1 + y3)/2)
- Using the given coordinates, the midpoint of AB is ((1 + 1)/2, (6 + 4)/2) = (1, 5) and the midpoint of AC is ((1 + 5)/2, (6 + 4)/2) = (3, 5).
2. Find the slopes of two perpendicular bisectors of the sides.
- The slope of the perpendicular bisector of side AB is the negative reciprocal of the slope of AB, which is (-6 - 4)/(1 - 1) = undefined (vertical line).
- The slope of the perpendicular bisector of side AC is the negative reciprocal of the slope of AC, which is (4 - 6)/(5 - 1) = -1/2.
3. Find the equations of the perpendicular bisectors using the midpoints and their respective slopes.
- The equation of the perpendicular bisector of AB passing through the midpoint (1, 5) is x = 1.
- The equation of the perpendicular bisector of AC passing through the midpoint (3, 5) is y = -1/2 * x + 7/2.
4. Find the intersection point of the perpendicular bisectors.
- Since the equation of the perpendicular bisector of AB is x = 1 and the equation of the perpendicular bisector of AC is y = -1/2 * x + 7/2, we can substitute x = 1 into the equation of the bisector of AC to find the y-coordinate of the circumcenter.
y = -1/2 * 1 + 7/2 = 3
- Therefore, the circumcenter of triangle ABC is (1, 3).