5. Find the vertices of the polygon after a dilation with a scale factor of 1/3. A(3,-3) B(-6,3) C(-9,9).

Answer: To find the vertices of the polygon after a dilation with a scale factor of 1/3, we have to multiply the coordinates of each vertex by the scale factor. Here, the scale factor is 1/3 which means each coordinate of the vertices will be multiplied by 1/3.

Let's find the new coordinates after dilation for each vertex:

For vertex A(3,-3): The new x-coordinate will be 3 * (1/3) = 1 The new y-coordinate will be -3 * (1/3) = -1 So the new vertex A' will be at (1,-1).

For vertex B(-6,3): The new x-coordinate will be -6 * (1/3) = -2 The new y-coordinate will be 3 * (1/3) = 1 So the new vertex B' will be at (-2,1).

For vertex C(-9,9): The new x-coordinate will be -9 * (1/3) = -3 The new y-coordinate will be 9 * (1/3) = 3 So the new vertex C' will be at (-3,3).

Therefore, after the dilation with a scale factor of 1/3, the vertices of the polygon will be A'(1,-1), B'(-2,1), and C'(-3,3).

Extra: Dilation is a transformation that changes the size of a figure without altering its shape. The scale factor determines how much the figure is enlarged or reduced. If the scale factor is greater than 1, the figure gets larger; if it is between 0 and 1, the figure gets smaller. If the scale factor is equal to 1, the figure remains the same size.

When we apply dilation to a polygon, we multiply the coordinates of each of its vertices by the scale factor to get the coordinates of the dilated polygon's vertices. This operation is performed independently on the x and y coordinates of each vertex.

A scale factor of 1/3, as used in this example, means the resulting polygon will be one-third of the size of the original polygon in every dimension. It is important to note that while the size of the figure changes, the angles in the polygon remain the same, preserving the shape. This property makes dilation a type of "similarity transformation," as the original and the dilated figures are similar to each other.