The discussion revolves around finding values for the slope (m) and y-intercept (b) of a line such that the points (8, 2) and (4, 8) are symmetric about this line. The conversation touches on the concepts of symmetry, perpendicular bisectors, and the relationship between points and lines in a geometric context.
Participants do not reach a consensus on the method for finding m and b, with differing views on the initial approach to the problem and the necessary geometric considerations.
There is an implicit assumption that understanding the geometric relationship between the points and the line is crucial, but specific mathematical steps and definitions remain unresolved.
Before you get to that, what does it mean for the points to be symmetric about y = mx + b?RTCNTC said:Determine values for m and b so that the points (8, 2) and (4, 8) are symmetric about the line y = mx + b.
Do I plug the coordinates of each point into the formula
y = mx + b individually to find values for m and b?
Absolutely NOT! That would give you the equation of the line that contains (8, 2) and (4, 6) but these two points do NOT lie on the line you seek! They are symmetric about that line. In particular, the line you seek must be the perpendicular bisector of the line segment between (8, 2) and (4, 8).RTCNTC said:Determine values for m and b so that the points (8, 2) and (4, 8) are symmetric about the line y = mx + b.
Do I plug the coordinates of each point into the formula
y = mx + b individually to find values for m and b?
HallsofIvy said:Absolutely NOT! That would give you the equation of the line that contains (8, 2) and (4, 6) but these two points do NOT lie on the line you seek! They are symmetric about that line. In particular, the line you seek must be the perpendicular bisector of the line segment between (8, 2) and (4, 8).
What are the coordinates of the point midway between (8, 2) and (4, 6)? What is the slope of the line through (8, 2) and (4, 6)? What is the slope of a line perpendicular to that line? Finally, what is the equation of the line through that midpoint perpendicular to that line?