For the semicircle above, point \(C\) represents the midpoint of \(\operatorname{arc} A B\). Which of the following represent the coordinates of point \(C\) ? A. \((6,7)\) B. \((7,7)\) C. \((4,6)\) D. \((7,6)\)

First, we need to find the center of the semicircle. The center will be the midpoint of the straight line segment AB connecting the two end points A and B. Since the problem doesn't give any coordinates for A or B, we'll assume that A is at (0, 0) and B is at (14, 0), as the distance between the two points must be 14 to align with given options. The midpoint of a line segment with end points (x1, y1) and (x2, y2) can be found using the formula: Midpoint = \((\frac{x1+x2}{2}, \frac{y1+y2}{2})\) So, we have: Midpoint = \((\frac{0+14}{2}, \frac{0+0}{2})\) = \((7, 0)\) Therefore, the center of the semicircle is at point (7, 0).

Now we need to find the radius of the semicircle. Since the distance between the end points A and B is 14, the diameter of the semicircle is also 14. The radius is half the diameter, so the radius of the semicircle is: Radius = \(\frac{14}{2}\) = \(7\)

Now that we have the center of the semicircle and its radius, we can determine the coordinates of point C. Since C is the midpoint of the arc, it must lie on the circle and be diametrically opposite to the center (7, 0). Using the property of a circle, we have: \((x-7)^2 + (y-0)^2 = 7^2\) or \((x-7)^2 + y^2 = 49\) Now we'll try inserting the coordinates given in the options A, B, C, and D to see which option satisfies the equation: A. \((6,7)\): \((6-7)^2 + (7)^2 = 1^2 + 7^2 = 1 + 49 = 50\) (doesn't satisfy) B. \((7,7)\): \((7-7)^2 + (7)^2 = 0^2 + 7^2 = 0 + 49 = 49\) (satisfies) C. \((4,6)\): \((4-7)^2 + (6)^2 = 3^2 + 6^2 = 9 + 36 = 45\) (doesn't satisfy) D. \((7,6)\): \((7-7)^2 + (6)^2 = 0^2 + 6^2 = 0 + 36 = 36\) (doesn't satisfy) Option B satisfies the equation and represents the coordinates of point C: \((7,7)\).