To find the coordinates of \( B' \) after translating point \( B = (4, 10) \) right by 12 and down by 19, you adjust the x-coordinate and y-coordinate accordingly.
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Translate right by 12: \[ x' = 4 + 12 = 16 \]
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Translate down by 19: \[ y' = 10 - 19 = -9 \]
Putting these together, the coordinates of \( B' \) are:
\[ B' = (16, -9) \]
So, the answer is: \[ B' = (16, -9) \]