Question

consider the line segment whose endpoints are the (0,24) and (6,0). Let line l be the perpendicular bisector of this line segment. If line l passes through (11,a) What is the value of a?

Answer 1

the midpoint of the line segment with endpoints (0, 24) and (6, 0) is the average of the coordinates:

midpoint = ((0 + 6) / 2, (24 + 0) / 2) = (3, 12)

now, to find the equation of the line perpendicular to this segment, we need to determine its slope, which will be the negative reciprocal of the slope of the original line.

original line slope: (0 - 24) / (6 - 0) = (-24) / 6 = -4

so, the slope of the perpendicular bisector line (line l) is 1/4 (the negative reciprocal of -4).

now, we have the midpoint (3, 12) and the slope (1/4) of the perpendicular bisector line. we can use the point-slope form of a line to find the equation:

y - y1 = m(x - x1)

where (x1, y1) is the midpoint (3, 12) and m is the slope (1/4).

y - 12 = (1/4)(x - 3)

now, let's find the value of a when x = 11:

y - 12 = (1/4)(11 - 3)

y - 12 = (1/4)(8)

y - 12 = 2

now, add 12 to both sides to find y:

y = 12 + 2
y = 14

so, when x = 11, the point (11, a) lies on the perpendicular bisector line, and a = 14. i tried my best to explain this well, so i hope you understand!