Question

C is translated horizontally to D, such that angle BDC is 90 degree

Answer 1

[AI-generated answer — Bot GPT 3.5]

The translation can be represented by the vectors between the corresponding points. If we denote the vector from C to D as vector CD, and the horizontal translation vector as vector T, then the translation from C to D is CD = C + T.

Given that angle BDC is 90 degrees, this means that the vectors CB and CD are perpendicular to each other. So, the dot product of these vectors must be zero:

CB ∙ CD = 0

Substitute the vectors in terms of their components:

(B - C) ∙ (D - C) = 0

Expand the dot product:

(Bx - Cx)(Dx - Cx) + (By - Cy)(Dy - Cy) = 0

Since the translation is horizontal, the y-coordinates of C and D are the same. Therefore, Cy = Dy. Additionally, D lies along the x-axis, so Dy = 0.

Therefore, the equation simplifies to:

(Bx - Cx)(Dx - Cx) = 0

Since Dx - Cx ≠ 0, this implies that Bx = Cx, and the x-coordinate of point B is the same as the x-coordinate of point C.

In conclusion, if C is translated horizontally to D such that angle BDC is 90 degrees, it means that B and C have the same x-coordinate.