MHB Finding Values for m and b to Create a Symmetric Line

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To find values for m and b that make the points (8, 2) and (4, 8) symmetric about the line y = mx + b, one must determine the perpendicular bisector of the segment connecting these points. Plugging the coordinates into the line equation y = mx + b is incorrect, as it does not yield the desired line of symmetry. Instead, the midpoint of the segment and the slope of the line connecting the two points should be calculated to find the slope of the perpendicular bisector. The final equation of the line can then be derived from this midpoint and slope. Understanding these geometric relationships is crucial for solving the problem correctly.
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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?
 
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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?
Before you get to that, what does it mean for the points to be symmetric about y = mx + b?

-Dan
 
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).

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?
 
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?

Thank you. I have been away from this site for more than 2 weeks.
 

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