Final answer:
The equation for the perpendicular bisector k of line segment PQ is y = 2x - 6, derived using the midpoint of PQ and the negative reciprocal slope of line PQ.
Step-by-step explanation:
To find the equation of the perpendicular bisector line k for the line segment with endpoints P(10,4) and Q(2,8), first, we need to calculate the midpoint of PQ. The midpoint M will be the average of the x-coordinates and the y-coordinates of P and Q. So, M = ((10+2)/2, (4+8)/2) = (6, 6).
Next, we find the slope of line segment PQ. The slope m of a line passing through points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1). Thus, with P(10,4) and Q(2,8), the slope of PQ is m = (8 - 4) / (2 - 10) = 4 / (-8) = -1/2. The slope of the perpendicular bisector, line k, is the negative reciprocal of the slope of PQ, which would be 2.
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Using the slope of line k and its point M(6,6), we can substitute into the equation to find b: 6 = 2(6) + b. Solving this equation gives us b = -6. Therefore, the linear equation for the perpendicular bisector k in slope-intercept form is y = 2x - 6.