Question

ΔABC has A (3, -7), orthocenter H (3, -1), the center of the circumcircle is I (-2, 0). Find the point C - coordinate that knows the positive x-coordinate.

a. C(3, 5)
b. C(-2, 5)
c. C(3, -5)
d. C(-2, -5)

Answer 1

The point C - coordinate with a positive x-coordinate can be found by using the fact that the orthocenter of a triangle is the point where the altitudes intersect. In this case, we can use the coordinates of A and H to find the slope of the altitude from A and determine the x-coordinate of C. The answer is option c. C(3, -5).

Step-by-step explanation:

To find the point C with a positive x-coordinate, we can use the fact that the orthocenter of a triangle is the point where the altitudes intersect. Let's use point A and the orthocenter H to find the slope of the altitude from point A. The slope of the altitude is the negative reciprocal of the slope of the line that passes through A and H. Using the coordinates of A (3, -7) and H (3, -1), we can calculate the slope of the line:
m = (-1 -(-7))/(3-3) = -6/0 = undefined.

Since the slope is undefined, we know that the altitude is a vertical line passing through the x-coordinate of A. Therefore, point C will have the same x-coordinate as A, which is c. C(3, -5), so the answer is option c.