Perpendicular Bisector
A line that is perpendicular to a segment at its midpoint.
A perpendicular bisector is a line or a ray that intersects a given line segment at its midpoint and at a right angle. In other words, it divides the line segment into two equal parts, and the two parts are mirrored images of each other with respect to the bisector.
For example, if we have a line segment AB, and we draw a perpendicular bisector PQ that intersects AB at its midpoint, the two parts of the line segment, AP and PB, will be of equal length and will be symmetric with respect to the bisector PQ.
Perpendicular bisectors have many practical applications in geometry, such as finding the center of a circle or the location of an earthquake. They can also be used to prove theorems and solve geometric problems.
To find the equation of a perpendicular bisector, we first need to find the midpoint of the given line segment AB, which can be done by taking the average of its endpoints:
midpoint M = (A + B)/2
Then, we can find the slope of the line segment AB:
slope of AB = (B – A)/(x_B – x_A)
where x_A and x_B are the x-coordinates of A and B, respectively.
The slope of the perpendicular bisector PQ is the negative reciprocal of the slope of AB:
slope of PQ = -1/slope of AB
Finally, we can use the point-slope form of the equation of a line to find the equation of PQ, given its slope and the midpoint M:
y – y_M = slope of PQ * (x – x_M)
where x_M and y_M are the coordinates of the midpoint M.
In summary, a perpendicular bisector is a line or ray that intersects a given line segment at its midpoint and at a right angle. It is useful in geometry for finding the center of a circle, solving geometric problems, and proving theorems. The equation of a perpendicular bisector can be found by finding the midpoint of the line segment, finding the slope of the line segment, finding the slope of the perpendicular bisector, and using the point-slope form of the equation.
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