To find the values of \( h \) and \( k \) in the functions \( f(x) = x + h \) and \( g(y) = y + k \), we need to understand how translations work on a coordinate grid.
When triangle \( ABC \) is translated to triangle \( A'B'C' \), each point's coordinates change based on the values of \( h \) and \( k \). Specifically:
- For a translation in the x-direction, the x-coordinates of points increase by \( h \). This means \( A' = (A_x + h, A_y) \), \( B' = (B_x + h, B_y) \), and \( C' = (C_x + h, C_y) \).
- For a translation in the y-direction, the y-coordinates of points increase by \( k \). This means \( A' = (A_x, A_y + k) \), \( B' = (B_x, B_y + k) \), and \( C' = (C_x, C_y + k) \).
Since the triangle is translated by both \( h \) and \( k \), the new coordinates of points \( A' \), \( B' \), and \( C' \) can be expressed as:
- \( A' = (A_x + h, A_y + k) \)
- \( B' = (B_x + h, B_y + k) \)
- \( C' = (C_x + h, C_y + k) \)
To find specific values for \( h \) and \( k \), you need the original coordinates of triangle \( ABC \) and the new coordinates of triangle \( A'B'C' \). The values \( h \) and \( k \) can be determined by:
\[ h = A'_x - A_x \] \[ k = A'_y - A_y \]
You would perform the same calculations for points \( B \) and \( C \) to confirm consistency.
Without the specific coordinates of triangles \( ABC \) and \( A'B'C' \), \( h \) and \( k \) cannot be determined. Please provide those coordinates or any additional information if available.
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