If ABCD is a square, X is the mid point of AB and Y is the mid point of BC, then which of the following is NOT correct?

Options

A. The triangles ADX and BAY are congruent
B. `angle DXA = angle AYB`
C. `angle ADX = angle BAY`
D. DX is not perpendicular to AY

Answer

To solve the problem, we need to analyze the given square ABCD and the midpoints X and Y. We will then check the statements regarding the triangles formed.

### Step-by-Step Solution:

1. **Identify the Square and Midpoints**:
- Let the vertices of the square ABCD be:
- A(0, 0)
- B(1, 0)
- C(1, 1)
- D(0, 1)
- The midpoint X of AB is:
- X = ((0 + 1)/2, (0 + 0)/2) = (0.5, 0)
- The midpoint Y of BC is:
- Y = ((1 + 1)/2, (0 + 1)/2) = (1, 0.5)

2. **Identify the Triangles**:
- We have two triangles to consider:
- Triangle AXY (with vertices A, X, Y)
- Triangle BXY (with vertices B, X, Y)

3. **Check the Congruence of Triangles**:
- For two triangles to be congruent, they must have:
- Three pairs of equal sides or
- Two pairs of equal sides and the included angle or
- Three pairs of equal angles.
- We will check the sides and angles of triangles AXY and BXY.

4. **Calculate the Lengths of the Sides**:
- Length of AX:
- AX = distance from A(0, 0) to X(0.5, 0) = 0.5
- Length of XY:
- XY = distance from X(0.5, 0) to Y(1, 0.5) = √((1 - 0.5)² + (0.5 - 0)²) = √(0.25 + 0.25) = √0.5
- Length of AY:
- AY = distance from A(0, 0) to Y(1, 0.5) = √((1 - 0)² + (0.5 - 0)²) = √(1 + 0.25) = √1.25

5. **Check the Angles**:
- Angle AXY and angle BXY can be calculated using the slopes of the lines:
- Slope of AX = (0 - 0)/(0.5 - 0) = 0
- Slope of AY = (0.5 - 0)/(1 - 0) = 0.5
- Since both triangles share the angle at X, we can compare the remaining angles.

6. **Conclusion**:
- After checking the sides and angles, we can conclude that the triangles AXY and BXY are not congruent.
- Therefore, the statement that claims they are congruent is NOT correct.

### Final Answer:
The statement that triangle AXY is congruent to triangle BXY is NOT correct.