To analyze the triangle DEF and its translated version D'E'F', we will start with the coordinates of the vertices of the original triangle DEF and then determine the coordinates of the transformed triangle D'E'F'.
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Coordinates of vertices of triangle DEF: Since DEF is an equilateral triangle with side length 8, we can conveniently place it in the coordinate system. One common way is to position the triangle so that one vertex is at the origin and the others lie symmetrically:
- Let \( D(0, 0) \) be one vertex.
- \( E(8, 0) \) is the second vertex, directly to the right of D.
- For the third vertex \( F \), we can calculate its position. The height \( h \) of the equilateral triangle can be found using the formula: \[ h = \frac{\sqrt{3}}{2} \times \text{side length} = \frac{\sqrt{3}}{2} \times 8 = 4\sqrt{3} \] Thus, \( F \) is located at \( \left(4, 4\sqrt{3}\right) \).
The coordinates of the vertices of DEF are:
- \( D(0, 0) \)
- \( E(8, 0) \)
- \( F(4, 4\sqrt{3}) \)
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Apply translation to get D'E'F': The translation mentioned is 9 units down and 2 units to the left. This means we will subtract 2 from the x-coordinates and 9 from the y-coordinates of each point.
Let's calculate the new positions:
- For \( D(0, 0) \): \[ D' = (0 - 2, 0 - 9) = (-2, -9) \]
- For \( E(8, 0) \): \[ E' = (8 - 2, 0 - 9) = (6, -9) \]
- For \( F(4, 4\sqrt{3}) \): \[ F' = (4 - 2, 4\sqrt{3} - 9) = (2, 4\sqrt{3} - 9) \]
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Final coordinates of triangle D'E'F':
- \( D'(-2, -9) \)
- \( E'(6, -9) \)
- \( F'\left(2, 4\sqrt{3} - 9\right) \)
Therefore, the coordinates of the vertices of the triangle after translation D'E'F' are:
- \( D'(-2, -9) \)
- \( E'(6, -9) \)
- \( F'\left(2, 4\sqrt{3} - 9\right) \)
This completes our translation of the triangle from DEF to D'E'F'.
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