Defining a Perpendicular Bisector
Any instance of dividing a line into two equal halves indicates that the line has been bisected. When bisecting a line into two equal or congruent halves at right angles, we always construct a perpendicular bisector. As the term suggests the perpendicular bisector is a line that intersects another line at right angles, or 90°. This intersection of the bisector and another line divides the original line into two congruent parts.
The image gives us a better idea of a perpendicular bisector.
In the image we observe that there are two lines AB and CD that intersect each other at the point M. The line CD in this case is the perpendicular bisector of the line AB. Two distinct aspects of the perpendicular bisector can be seen here:
- Lines AM and BM are equal in length,AM = BM
- Angles CMA and CMB are right angles,∠CMA =∠CMB = 90°
[Note:Another feature of the perpendicular bisector that will be introduced in higher grades is that every point on a perpendicular bisector is equidistant from the points A and B]
Constructing a Perpendicular Bisector
A perpendicular bisector of a line can be constructed in three simple steps with the help of a compass and a ruler.
Step 1:
Draw a line PQ with the help of a ruler. With a radius of more than half of the length of the line PQ, and with P as the center, draw two arcs above and below the line PQ.
Step 2:
With Q as the center and with the same radius, draw two arcs that intersect the first two arcs, above and below the line PQ. Mark the points of intersection as X and Y.
Step 3:
Using a ruler, draw a line connecting the points X and Y, and this line intersects the line PQ at J.
So we can observe that line XY is the perpendicular bisector of line PQ. In this case,
JP = JQ
∠PJX =∠QJX = 90°
Rapid Recall
Solved Examples
Example 1:
Jason has constructed a perpendicular bisector QR to a line MN. He wants to know the measure of the angles formed at point A, as well as whether MA is equal to AN. Help Jason find the answers.
Solution:
Stated in the question:
Jason has constructed a perpendicular bisector QR, that intersects the line MN at A.
A perpendicular bisector intersects a given line and divides it into two equal halves. The angles formed at the point of intersection are right angles.
Hence, Jason finds that, MA = AN and ∠QAM = ∠QAN =90°
Example 2:
State whether the following statements are true or false.
- Parallel lines intersect to result in a perpendicular bisector
- Four 60° angles are formed when a perpendicular bisector of a line is drawn.
- A perpendicular bisector divides a line into two equal parts
Solution:
- Parallel lines intersect to result in a perpendicular bisector – False – This is because parallel lines do not intersect at all, hence a perpendicular bisector is not possible in the case of parallel lines.
- Four 60° angles are formed when a line bisects another line – False – This is because a perpendicular bisector intersects a line at right angles that are 90°.
- A perpendicular bisector divides a line into two equal parts – True – This is because a perpendicular bisector always divides a line into two congruent parts or halves.
In geometry, the term congruent indicates that a line is equal in length to another line. In terms of shapes, one shape is congruent to another if both shapes can be placed precisely over each other.
We use an instrument called a protractor to measure angles in geometry.
A perpendicular bisector divides a line into two congruent parts and the bisector also forms right angles at the point of intersection with the original line.