To find the equation of the mirror line for the segment with endpoints (5, 4) and (-1, -1), you can first find the slope of the line segment. The slope is calculated using the formula:
slope = (y2 - y1) / (x2 - x1)
Substituting the coordinates of the two points into the formula gives:
slope = (-1 - 4) / (-1 - 5) = -5 / -6 = 5/6
To find the equation of the mirror line, you can use the slope of the line segment and the midpoint of the line segment. The midpoint is calculated using the formula:
midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Substituting the coordinates of the two points into the formula gives:
midpoint = ((5 + -1) / 2, (4 + -1) / 2) = (2, 1.5)
The equation of the mirror line is perpendicular to the line segment, so its slope is the negative reciprocal of the slope of the line segment. The negative reciprocal of 5/6 is -6/5. To find the equation of the mirror line, you can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. Substituting the slope and midpoint into the equation gives:
y = (-6/5)x + b
Substituting the y-coordinate of the midpoint (1.5) for y gives:
1.5 = (-6/5) * 2 + b
3 = -6 + 5b
9 = 5b
b = 1.8
Therefore, the equation of the mirror line is y = (-6/5)x +