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

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SUMMARY

The discussion focuses on determining the values of m and b for the line equation y = mx + b such that the points (8, 2) and (4, 8) are symmetric about this line. It is established that the line must be the perpendicular bisector of the segment connecting these two points. The midpoint of the segment is calculated, and the slopes of the line segment and its perpendicular bisector are discussed to derive the equation of the desired line.

PREREQUISITES
  • Understanding of linear equations, specifically the form y = mx + b.
  • Knowledge of the concept of symmetry in geometry.
  • Ability to calculate midpoints between two points.
  • Familiarity with slopes and perpendicular lines in coordinate geometry.
NEXT STEPS
  • Learn how to calculate the midpoint of a line segment in coordinate geometry.
  • Study the properties of perpendicular bisectors in relation to symmetry.
  • Explore the derivation of line equations from given points and slopes.
  • Investigate the geometric interpretation of linear equations and their graphs.
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Students and educators in mathematics, particularly those focusing on geometry and algebra, as well as anyone interested in understanding the concepts of symmetry and line equations.

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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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