Let D be the mid-point of a straight line AB and let C be a point different from D such that AC=BC, then.

Options

A. `AC bot AB`
B. `angle BDC=90^@`
C. `angle BDC` is acute
D. `angle BDCgt90^@`

Answer

To solve the problem step by step, we will analyze the given information and use geometric properties to arrive at the conclusion.

### Step 1: Identify the points and their relationships
Let A and B be two points on a straight line, with D being the midpoint of line segment AB. This means that AD = DB.

**Hint:** Remember that the midpoint divides the segment into two equal parts.

### Step 2: Place point C
Point C is given such that AC = BC. This means that point C is equidistant from points A and B.

**Hint:** Visualize point C as being located somewhere above or below the line segment AB, forming two equal lengths AC and BC.

### Step 3: Forming triangle ABC
Since AC = BC, triangle ABC is isosceles. In fact, since D is the midpoint of AB, triangle ABC can be shown to be equilateral if we can prove that all sides are equal.

**Hint:** Consider the properties of isosceles triangles and how they relate to the angles.

### Step 4: Establishing the triangle's angles
In triangle ABC, since AC = BC, the angles opposite these sides are equal. Let’s denote the angle at A as ∠CAB and the angle at B as ∠CBA. Therefore, ∠CAB = ∠CBA.

**Hint:** Use the fact that the sum of angles in a triangle is 180 degrees.

### Step 5: Analyzing the angles
Since D is the midpoint and AC = BC, we can conclude that triangle ABC is not only isosceles but also equilateral. Thus, each angle in triangle ABC measures 60 degrees.

**Hint:** Recall that in an equilateral triangle, all angles are equal and measure 60 degrees.

### Step 6: Investigating angle BDC
Now, we need to examine angle BDC. Since D is the midpoint of AB and C is above the line segment, the line segment CD acts as both a median and an altitude in triangle ABC.

**Hint:** In an equilateral triangle, the median from a vertex to the opposite side is also the altitude.

### Step 7: Concluding angle BDC
Since CD is both a median and an altitude, angle BDC is a right angle (90 degrees).

**Hint:** Remember that the altitude in a triangle creates a right angle with the base.

### Final Conclusion
Thus, we conclude that angle BDC = 90 degrees.

### Summary of Steps
1. Identify points A, B, and D.
2. Place point C such that AC = BC.
3. Form triangle ABC and analyze its properties.
4. Establish the angles in triangle ABC.
5. Investigate angle BDC using properties of medians and altitudes.
6. Conclude that angle BDC = 90 degrees.