Understanding Segment Bisectors: Definition, Examples, and Equation of a Bisector

Segment bisector

A segment bisector is a line, ray, or segment that divides a segment into two equal parts, also known as halves

A segment bisector is a line, ray, or segment that divides a segment into two equal parts, also known as halves. In other words, it is a line that passes through the midpoint of a segment, dividing it into two congruent segments.

To understand how a segment bisector works, let’s consider an example:
Let’s say we have a line segment AB. The midpoint of this line segment is a point M, which is equidistant from both points A and B.

Now, if we draw a line passing through point M and intersecting AB, that line is called a segment bisector. The bisector divides the segment AB into two equal halves. This means that the length of segment AM is equal to the length of segment MB.

To find the equation of a segment bisector, we need to determine the coordinates of the midpoint and use those coordinates to create an equation for the bisector. We can use various methods, such as the midpoint formula or symmetry, to find the midpoint of a segment.

For example, let’s find the equation of the bisector for the segment with endpoints A(-3, 2) and B(5, 6).

Step 1: Find the coordinates of the midpoint using the midpoint formula:
Midpoint formula: ( (x1 + x2) / 2, (y1 + y2) / 2 )
Midpoint = ((-3 + 5) / 2, (2 + 6) / 2)
Midpoint = (2 / 2, 8 / 2)
Midpoint = (1, 4)

Step 2: Determine the slope of the original segment.
Slope formula: ( (y2 – y1) / (x2 – x1) )
Slope = (6 – 2) / (5 – (-3))
Slope = 4 / 8
Slope = 1/2

Step 3: Determine the negative reciprocal of the original slope. This will give us the slope of the bisector.
Negative reciprocal of 1/2 is -2/1 or -2.

Step 4: Use the point-slope form of a line to find the equation of the bisector.
Point-slope form: (y – y1) = m(x – x1)
Using the midpoint (1, 4) and the slope -2, we get:
(y – 4) = -2(x – 1)

Simplifying this equation gives us the equation of the segment bisector:
y – 4 = -2x + 2

This is the equation of the bisector for the segment with endpoints A(-3, 2) and B(5, 6).

Remember that a segment bisector divides a segment into two equal halves, and its equation can be found using the midpoint of the segment and the slope of the original segment.

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