Question

Given vertices H(-10,7), J(-6,3) and K(-2,3) find the circumcenter of the triangle

Answer 1

For triangle with vertices H(-10, 7), J(-6, 3), K(-2, 3), circumcenter O is (-4, 9) by solving distance equations.

Consider the triangle with vertices H(-10, 7), J(-6, 3), and K(-2, 3). To find the circumcenter O, equate the distances from O to each vertex using the distance formula.

Let the coordinates of O be (x, y).

1. OH = OJ:

`\((x + 10)^2 + (y - 7)^2 = (x + 6)^2 + (y - 3)^2\)`

Simplifying, 8x - 8y + 104 = 0, or x - y + 13 = 0 (Equation 1).

2. OH = OK:

`\((x + 10)^2 + (y - 7)^2 = (x + 2)^2 + (y - 3)^2\)`

Simplifying, 16x - 8y + 136 = 0, or 2x - y + 17 = 0 (Equation 2).

Solving Equations 1 and 2, we find x = -4 and y = 9.

Therefore, the circumcenter O has coordinates (-4, 9).