Question

The city planner wants to put recycling bins at the circumcenter of the triangular park. What are the coordinates of the recycling bins?

a) (3, 1)
b) (0, 1)
c) (-3, 1)
d) (3, 2)

Answer 1

To find the coordinates of the recycling bins at the circumcenter of a triangle, you need to find the intersection point of the perpendicular bisectors of the triangle's sides.

Step-by-step explanation:

The coordinates of the recycling bins at the circumcenter of a triangle can be found by finding the intersection point of the perpendicular bisectors of the sides of the triangle. Let's assume the coordinates of the triangle vertices are A(x1, y1), B(x2, y2), and C(x3, y3).

1. Find the midpoints of the sides of the triangle using the midpoint formula: Midpoint of AB is M((x1+x2)/2, (y1+y2)/2), Midpoint of AC is N((x1+x3)/2, (y1+y3)/2), and Midpoint of BC is O((x2+x3)/2, (y2+y3)/2).
2. Calculate the slopes of the sides of the triangle using the slope formula: The slope of AB is m1 = (y2-y1)/(x2-x1), the slope of AC is m2 = (y3-y1)/(x3-x1), and the slope of BC is m3 = (y3-y2)/(x3-x2).
3. Find the equations of the perpendicular bisectors of the sides of the triangle using the point-slope form of a line. For example, the equation of the perpendicular bisector of AB is given by y - (y1+y2)/2 = (-1/m1)(x - (x1+x2)/2).
4. Find the intersection point of the perpendicular bisectors. Solve the system of equations formed by the equations of the perpendicular bisectors to find the coordinates of the circumcenter.

After calculating the coordinates of the circumcenter, you can determine which answer choice matches the coordinates.