Analyse whether lies on the perpendicular bisector of .

Question

Homi · Accepted Answer

Understand the Question

A point lies on the perpendicular bisector of a line segment if and only if it is equidistant from the two endpoints of that segment.
To determine whether point $A$ lies on the perpendicular bisector of segment $BC$ , we need to prove whether $AB = AC$ using the geometric properties shown in the diagram.

Step 1 · Identify Geometric Properties from the Diagram

Diagram 1

From the given figure:

$BB' \parallel CC'$
Line $l$ is perpendicular to both $BB'$ and $CC'$ .
The tick marks indicate that line $l$ bisects both $BB'$ and $CC'$ .
Therefore, line $l$ is the perpendicular bisector of both $BB'$ and $CC'$ , and point $A$ lies on line $l$ .

Step 2 · Apply Perpendicular Bisector Properties

Any point on a perpendicular bisector is equidistant from the segment's endpoints.

Since $A$ lies on the perpendicular bisector of $BB'$ : $AB = AB'$

Since $A$ lies on the perpendicular bisector of $CC'$ : $AC = AC'$

Step 3 · Use Properties of Quadrilateral $B'BCC'$

Since $BB' \parallel CC'$ and both are perpendicular to line $l$ , quadrilateral $B'BCC'$ is a rectangle.

In a rectangle, the diagonals are equal and bisect each other at point $A$ . Therefore: $AB = AC'$ $AB' = AC$

Step 4 · Conclude for the Perpendicular Bisector of $BC$

Combining the relations: $AB = AB' \quad \text{and} \quad AB' = AC$ $\implies AB = AC$

Since $A$ is equidistant from $B$ and $C$ ( $AB = AC$ ), point $A$ lies on the perpendicular bisector of segment $BC$ .

Answer

Yes, $A$ lies on the perpendicular bisector of $BC$ .

Common Mistakes

Overlooking the Equidistance Theorem: Forgetting that showing a point lies on a segment's perpendicular bisector only requires proving it is equidistant from both endpoints ( $AB = AC$ ).
Confusing Diagonal Bisectors: Assuming $A$ bisects the segments without establishing that quadrilateral $B'BCC'$ is a rectangle.

Question 12