Question

If the reflection of a point \( A(2,3) \) in the X-axis is \( B \); the reflection of \( B \) in the line \( x + y = 0 \) is \( C \) and the reflection of \( C \) in \( x - y = 0 \) is \( D \), then the point of intersection of the lines \( CD, AB \) is:

• A. \( (3, -2) \)
• B. \( (0,1) \)
• C. \( (4, -3) \)
• D. \( (2, -1) \)

Hint:

Reflections over axes and lines follow specific transformation rules. To find intersections, solve the system of equations formed by the given lines.

Answer 1

The Correct Option is D

Step 1: Find the Reflection of \( A(2,3) \) in the X-Axis The reflection of a point \( (x,y) \) in the X-axis is given by: \[ (x, -y). \] Thus, the reflection of \( A(2,3) \) is: \[ B(2,-3). \] Step 2: Find the Reflection of \( B(2,-3) \) in the Line \( x+y=0 \) The formula for reflecting a point \( (x,y) \) in the line \( x+y=0 \) is: \[ (x', y') = (-y, -x). \] Applying this: \[ C(-(-3), -2) = (3,-2). \] Step 3: Find the Reflection of \( C(3,-2) \) in the Line \( x-y=0 \) The formula for reflecting a point \( (x,y) \) in the line \( x-y=0 \) is: \[ (x', y') = (y, x). \] Applying this: \[ D(-2, 3). \] Step 4: Find the Intersection of \( CD \) and \( AB \) The equation of line \( AB \) passing through \( A(2,3) \) and \( B(2,-3) \) is: \[ x = 2. \] The equation of line \( CD \) passing through \( C(3,-2) \) and \( D(-2,3) \) has slope: \[ m = \frac{3 - (-2)}{-2 - 3} = \frac{5}{-5} = -1. \] Equation of line using point-slope form: \[ y + 2 = -1(x - 3). \] \[ y = -x + 1. \] Solving for \( x = 2 \): \[ y = -2 + 1 = -1. \] Thus, the intersection point is: \[ (2,-1). \] Step 5: Conclusion
Thus, the correct answer is \( \mathbf{(2,-1)} \).