Perpendicular Bisector Theorem
When a line divides another line segment into two equal halves through its midpoint at 90º, it is called the perpendicular of that line segment. The perpendicular bisector theorem states that any point on the perpendicular bisector is equidistant from both the endpoints of the line segment on which it is drawn. If a pillar is standing at the center of a bridge at an angle, all the points on the pillar will be equidistant from the end points of the bridge.
What is a Perpendicular Bisector?
A perpendicular bisector is a line segment that intersects another line segment at a right angle and it divides that other line into two equal parts at its midpoint.
What is Perpendicular Bisector Theorem?
The perpendicular bisector theorem states that any point on the perpendicular bisector is equidistant from both the endpoints of the line segment on which it is drawn.
In the above figure,
MT = NT
MS = NS
MR = NR
MQ = NQ
What is the Converse of Perpendicular Bisector Theorem?
The converse of the perpendicular bisector theorem states that if a point is equidistant from both the endpoints of the line segment in the same plane, then that point is on the perpendicular bisector of the line segment.
In the above image, XZ=YZ
It implies ZO is the perpendicular bisector of the line segment XY.
Proof of Perpendicular Bisector Theorem
Let us look at the proof of the above two theorems on a perpendicular bisector.
Perpendicular Bisector Theorem Proof
Consider the following figure, in which C is an arbitrary point on the perpendicular bisector of AB (which intersects AB at D):
Compare \(\Delta ACD\) and \(\Delta BCD\). We have:
- AD = BD
- CD = CD (common)
- ∠ADC =∠BDC = 90°
We see that \(\Delta ACD \cong \Delta BCD\) by the SAS congruence criterion. CA = CB,which means that C is equidistant from A and B.
Note: Refer to the SAS congruence criterion to understand why \(\Delta ACD\) and \(\Delta BCD\) are congruent.
Perpendicular Bisector Theorem Converse Proof
Consider CA = CB in the above figure.
To prove that AD = BD.
Draw a perpendicular line from point C that intersects line segment AB at point D.
Now, compare \(\Delta ACD\) and \(\Delta BCD\). We have:
- AC= BC
- CD = CD(common)
- ∠ADC = ∠BDC = 90°
We see that \(\Delta ACD \cong \Delta BCD\) by the SAS congruence criterion. Thus, AD = BD, which means that C is equidistant from A and B.
Important Notes
- The perpendicular bisector theorem and its converse can be proved by the SAS congruency criterion.
- The perpendicular bisector theorem is used in the construction of buildings, bridges, etc., and in making designs where we need to build something in the center and at equal distance from the endpoints.
Related Topics on Perpendicular Bisector Theorem
- Angle Bisector
- Constructing an angle of 90 degrees
- Congruent Triangles
- How do You Know if Two Line Segments are Perpendicular?
Solved Examples on Perpendicular Bisector Theorem
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Example 1: In a pyramid, line segment AD is the perpendicular bisector of triangle ABC on line segment BC. If AB = 20 feet and BD= 7 feet, find the length of side AC.
Solution
It is given that AD is the perpendicular bisector on the line segment BC.
So, By Perpendicular Bisector Theorem, any point on line segment AD is at an equal distance from points B and C. It implies, AB = AC
AC = 20 feet.
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Example 2: In any equilateral or isosceles triangle, can we say that the vertex between equal sides lies on the perpendicular bisector of the base?
SolutionDraw a perpendicular from vertex X that intersects segment YZ at point \(O\).
If XY =XZ,
Then, by converse of perpendicular bisector theorem, it is proved that OY=OZ .
\(\therefore\) Vertex X lies on the perpendicular bisector of the base YZ of triangle.
Frequently Asked Questions(FAQs)
What is the Perpendicular Bisector Theorem?
The perpendicular bisector theorem states that any point on the perpendicular bisector is equidistant from both the endpoints of the line segment on which it is drawn.
What is the Angle Bisector Theorem?
The angle bisector theorem states that in a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle.
What is an Example of a Perpendicular Bisector?
The median of a triangle is the line that joins the vertex of the triangle to the midpoint of the opposite side of the vertex. The median of an equilateral triangle is an example of a perpendicular bisector.
What is the Linear Pair Perpendicular Theorem?
The linear pair perpendicular theorem states that if two straight lines intersect at a point and the linear pair of angles they form have an equal measure, then the two lines are perpendicular to each other.
What is the Median of a Triangle?
The median of a triangle is a line segment which joins a vertex to the midpoint of the opposite side, thus bisecting that particular side. Every triangle has three medians which start from each vertex and intersect each other at the centroid of the triangle.