Question

With A and B as centres and any radius greater than half of AB, draw arcs on either side of AB so that they meet in X and Y as shown. Join XY

These are the two statements:
Statement 1: AB is the perpendicular bisector of XY
Statement 2: XY is the perpendicular bisector of AB
Choose the correct option from below.

• A. Only Statement 1 is true
• B. Only Statement 2 is true
• C. Both the statements are true
• D. None of the statements are true

Answer 1

The correct option is C

Both the statements are true

From the given question we can say that XY is the perpendicular bisector of AB. So, $\angle$ AOX = $\angle$ AOY = ${90}^{\circ }$ and AO = OB. So, statement 2 is true.

If AB is a perpendicular bisector of line XY, it should divide XY in such a way that XO = OY and $\angle$ XOA = $\angle$ AOY = ${90}^{\circ }$.

$\because \angle AOX=\angle YOA$

$\Rightarrow \angle XOA={90}^{\circ }$ similarly $\angle YOA={90}^{\circ }$---------(1)

Join AX, AY and BX, BY.
Consider $△$ AOX and $△$ AOY,

AX = AY (same radius)

AO = AO (common side)
$△$ AOX $\cong$ $△$ AOY (By S.S.S congruency)
Hence OX = OY-----------(2)
$\therefore$From (1) and (2) we can say that AB is perpendicular bisector of XY.

So statement 1 is also true.