Question

perpendicular bisector theorem explain in easy pls ans it ​

Answer 1

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.

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.

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

ΔACD and ΔBCD. We have:

• AD = BD
• CD = CD (common)
• ∠ADC =∠BDC = 90°

We see that ΔACD≅ΔBCD 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

ΔACD and ΔBCD 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 ΔACD and ΔBCD. We have:

• AC= BC
• CD = CD(common)
• ∠ADC = ∠BDC = 90°

We see that ΔACD≅ΔBCD 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.

Answer 2

Answer:

Perpendicular Bisector is a line or a segment perpendicular to a segment that passes through the midpoint of the segment. In the figure, ↔AB is a perpendicular bisector of ¯CD . Any point on the perpendicular bisector is equidistant from the endpoints of the line segment.

Step-by-step explanation:

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.

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