Questions: A hexagon is graphed on a coordinate grid and then dilated by a scale factor of 'w' with the origin as the center of dilation. If a vertex of the original hexagon was located at (8,-2), which ordered pair represents the vertex of the new hexagon after the dilation? 4. (8/w, -2/w) 4) (8+w,-2+w) 4] (8w,-2w) 4) (8+1/w,-2+1/w)

Solution Steps

To solve this problem, we need to understand the concept of dilation in geometry. Dilation involves resizing a shape by a scale factor while keeping the origin as the center of dilation. The coordinates of any point after dilation can be found by multiplying the original coordinates by the scale factor.

1. Identify the original coordinates of the vertex, which are (8, -2).
2. Multiply each coordinate by the scale factor \( w \) to get the new coordinates.

We are given a hexagon that is dilated by a scale factor \( w \) with the origin as the center of dilation. A vertex of the original hexagon is located at \( (8, -2) \). We need to determine the coordinates of this vertex after the dilation.

The formula for dilation with the origin as the center is: \[ (x', y') = (kx, ky) \] where \( k \) is the scale factor, and \( (x, y) \) are the original coordinates.

Here, \( k = w \), \( x = 8 \), and \( y = -2 \). Substituting these values into the dilation formula, we get: \[ (x', y') = (w \cdot 8, w \cdot (-2)) = (8w, -2w) \]

We compare the calculated coordinates \( (8w, -2w) \) with the given multiple-choice options:

1. \(\left(\frac{8}{w}, \frac{-2}{w}\right)\)
2. \((8 + w, -2 + w)\)
3. \((8w, -2w)\)
4. \(\left(8 + \frac{1}{w}, -2 + \frac{1}{w}\right)\)

The correct option is: \[ (8w, -2w) \]

Final Answer