Final Answer:
C'' is the result of reflecting point C across both the x-axis and the line y=-x. In coordinates, C'' is (2, 7) after these two reflections.
Explaination:
To reflect a point across the x-axis, we change the sign of its y-coordinate. Therefore, the reflected point C' is (2, -7).
Next, we reflect C' across the line y = -x. To do this, we find the slope of this line, which is -1. We then find the y-intercept by setting x equal to 0 and solving for y: y = -x, so y = 0 when x = 0. Therefore, the equation of the line is y = -x.
To reflect a point across a line with equation y = mx + b, we find the perpendicular bisector of the line segment connecting the point to its reflection across the x-axis. The slope of this bisector is the negative reciprocal of m. We then find the intersection point of this bisector and the line y = mx + b. The coordinates of this intersection point are the coordinates of the reflection across the line.
In our case, the slope of the bisector is -1/(-1) = 1. The equation of this bisector is y = x. The intersection point of this bisector and the line y = -x is (0, 0). Therefore, C'' (2, 7) reflects across both the x-axis and the line y = -x to (2, 7) reflects to (2, -7), which then reflects to (0, 0). So C'' is (0, 0).