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In the coordinate plane, the point A(3,4) is translated to the point A(6,2). Under the same translation, the points B(1,6) and C(5,1) are translated to B and C, respectively. What are the coordinates of B and C.

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Answer:

The coordinates of B' and C' are [tex]B'(x,y) = (4,4)[/tex] and [tex]C'(x,y) = (8, -1)[/tex].

Step-by-step explanation:

Vectorially speaking, the translation of a point is represented by the following operation:

[tex]P'(x,y) = P(x,y) + T(x,y)[/tex] (1)

Where:

[tex]P(x,y)[/tex] - Original point.

[tex]P'(x,y)[/tex] - Translated point.

[tex]T(x,y)[/tex] - Translation vector.

First, we need to calculate the translation vector after knowing that [tex]A(x,y) = (3,4)[/tex] and [tex]A'(x,y) = (6,2)[/tex]. That is:

[tex]T(x,y) = A'(x,y) - A(x,y)[/tex]

[tex]T(x,y) = (6,2) - (3,4)[/tex]

[tex]T(x,y) = (3, -2)[/tex]

Finally, we determine the coordinates of points B' and C':

[tex]B(x,y) = (1,6)[/tex], [tex]T(x,y) = (3, -2)[/tex]

[tex]B'(x,y) = (1,6) + (3,-2)[/tex]

[tex]B'(x,y) = (4,4)[/tex]

[tex]C(x,y) = (5,1)[/tex], [tex]T(x,y) = (3, -2)[/tex]

[tex]C'(x,y) = (5,1) + (3,-2)[/tex]

[tex]C'(x,y) = (8, -1)[/tex]

The coordinates of B' and C' are [tex]B'(x,y) = (4,4)[/tex] and [tex]C'(x,y) = (8, -1)[/tex].

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