A is the orthocentre of `DeltaABC` and D is reflection point of A w.r.t. perpendicualr bisector of BC, then orthocenter of `DeltaDBC` is :

Options

A. D
B. C
C. B
D. A

Answer

To solve the problem, we need to find the orthocenter of triangle DBC, given that A is the orthocenter of triangle ABC and D is the reflection of A with respect to the perpendicular bisector of side BC.

### Step-by-Step Solution:

1. **Identify the Properties of Triangle ABC**:
Since A is the orthocenter of triangle ABC, it implies that triangle ABC is a right triangle with the right angle at A.

**Hint**: Remember that the orthocenter of a triangle is the point where the altitudes intersect, and in a right triangle, the orthocenter is at the vertex of the right angle.

2. **Draw the Circle**:
Draw a circle with BC as the diameter. Since triangle ABC is right-angled at A, the circle will pass through points A, B, and C.

**Hint**: The property of a circle states that an angle subtended by a diameter is always a right angle.

3. **Draw the Perpendicular Bisector of BC**:
The perpendicular bisector of BC will also be a diameter of the circle. Let’s denote this diameter as PQ.

**Hint**: The perpendicular bisector of a chord in a circle passes through the center of the circle.

4. **Locate Point D**:
Point D is the reflection of point A with respect to the perpendicular bisector PQ. To find D, you can visualize or draw a line from A to PQ and extend it an equal distance on the other side of PQ.

**Hint**: The reflection of a point across a line can be found by measuring the perpendicular distance to the line and extending that distance on the opposite side.

5. **Determine the Orthocenter of Triangle DBC**:
Since D lies on the circle and BC is the diameter of the circle, angle BDC is a right angle (90 degrees). Therefore, triangle DBC is a right triangle with the right angle at D.

**Hint**: In a right triangle, the orthocenter is located at the vertex where the right angle is formed.

6. **Conclusion**:
Thus, the orthocenter of triangle DBC is point D itself.

**Final Answer**: The orthocenter of triangle DBC is D.