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 in D and C. Then he marks two more arcs with centres as C and D and radius more than $\frac{1}{2}CD$ as shown (intersecting in Y) but has no idea why.
Select the statement which explains him the reason.

• A. Because of equal radius, DY = CY and OD = OC. So $△$DOY is congurent to $△$COY
• B. Since OC = OD, $△$ODC in an isosceles triangle, base angles are equal.
• C. Because of equal radius, DY = CY and OD = DC. So $△$DOY is congurent to $△$COY.
• D. Since OC = OB, $△$ODC in an isosceles triangle, base angles are equal.

Answer 1

The correct option is A Because of equal radius, DY = CY and OD = OC. So $△$DOY is congurent to $△$COY

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 by SSS congruency criterion, $△$DOY is congruent to $△$COY.
So, corresponding parts of congruent triangles are equal which means, $\angle$ DOY = $\angle$COY and OY bisects the angle.
Hence, statement A is the one which gives the correct reason for OY being the angular bisector.