Question

Hari was asked to bisect a given angle, $\angle$ BOA as shown. He has done the following steps:
He draws an arc with centre O and some radius such that it cuts OB and OA at D and C respectively. Then he draws two more arcs with centres as C and D and radius more than $\frac{1}{2}$CD as shown (intersecting at Y) but has no idea why.

Select the correct statement which explains to him the reason for his construction.

Statement A : Because of equal radius, DY = CY and OD = OC. So $△$ DOY is congruent to $△$ COY

Statement B : Since OC = OD, $△$ ODC in an isosceles triangle, base angles are equal.

• A. Statements A and B are true, only Statement A explains the reason.
• B. Statements A and B are true, only Statement B explains the reason.
• C. Statements A and B are true, but none explains the reason.
• D. Statements A and B are false.

Answer 1

The correct option is A Statements A and B are true, only Statement A explains the reason.

Since we take the same radius while marking the arcs from D and C, it is obvious that DY = CY.

Also, when we take O as the centre and any radius, we mark D and C so that OD = OC.

Now if we consider, OY is the side common to both of them.

So, $△$ DOY $\cong$ $△$ COY [by SSS congruency criterion]
$\Rightarrow$ $\angle$ DOY = $\angle$ COY [CPCT]

Hence, statement A is the one which gives the correct reason for OY being the angle bisector.