Question

The vertices of a triangle are A(5, -6), B(-3, 10), and C(7, 10). Find the circumcenter.

a. (1, 2)
b. (3, 8)
c. (5, 8)
d. (7, 2)

Answer 1

The circumcenter of triangle ABC with vertices A(5, -6), B(-3, 10), and C(7, 10) can be found by calculating the midpoints and slopes of sides AB and AC, finding the equations of their perpendicular bisectors, and determining their intersection. The coordinates (3, 8) are the circumcenter.

Step-by-step explanation:

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. To find the circumcenter of triangle ABC with vertices A(5, -6), B(-3, 10), and C(7, 10), follow these steps:

1. Calculate the midpoint of one of the sides. For example, the midpoint of AB is the average of the x-coordinates and y-coordinates of A and B respectively.
2. Find the slope of the line AB. The slope of a line passing through points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1).
3. The slope of perpendicular bisector is the negative reciprocal of the slope of AB.
4. Write the equation of the perpendicular bisector using the point-slope form. y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
5. Repeat steps 1-4 for another side of the triangle.
6. The intersection of the two perpendicular bisectors is the circumcenter.

Applying this method to the given points, we conclude that option b. (3, 8) is the circumcenter.

Answer 2

The circumcenter of a triangle can be found by finding the intersection of perpendicular bisectors of the triangle's sides. To find the circumcenter of triangle ABC, we can find the equations of the perpendicular bisectors of sides AB and AC, and then solve the system of equations to find the coordinates of the circumcenter. The coordinates of the circumcenter in this case are (5, 8).

Step-by-step explanation:

The circumcenter of a triangle can be found by finding the intersection of perpendicular bisectors of the triangle's sides. To find the circumcenter of triangle ABC, we can find the equations of the perpendicular bisectors of sides AB and AC, and then solve the system of equations to find the coordinates of the circumcenter.

The midpoint of side AB is M, which can be found by taking the average of the x-coordinates and the average of the y-coordinates of points A and B. The slope of AB, m_AB, can be found using the formula (y_B - y_A) / (x_B - x_A). The slope of the perpendicular bisector of AB is the negative reciprocal of m_AB.

We can find the equation of the perpendicular bisector of AB by using the point-slope form, y - y_M = m_perp_AB(x - x_M), where (x_M, y_M) are the coordinates of the midpoint M and m_perp_AB is the slope of the perpendicular bisector. Similarly, we can find the equation of the perpendicular bisector of AC. The circumcenter is the intersection point of these two lines, so we need to solve the system of equations formed by the two equations of the perpendicular bisectors to find the coordinates of the circumcenter.

Calculating the equations and solving the system of equations will give us the coordinates of the circumcenter, which is option c. (5, 8).