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Draw the image of

of

's vertices and connect them.
Hint #22 / 4
For example, point

is

units to the left of

and

units up, so its image is


unit to the left and


units up from

.
Triangle A B C. Point B is six units to the right of point P and three units down, Point C is three units left from point P and up six units, and Point A is six units left of point P. There is a dashed line to highlight horizontally the three units left and vertically six units up from point P. There is also a dashed line to highlight horizontally one unit to the left and vertically two units up from point P. There is another plotted point that is one unit left and two units up from Point P.




Hint #33 / 4
Let's do the same thing with the remaining vertices.
Since our scale factor is


, every side length in

will be


the length of the corresponding side in

.
Another way to think about it is that since our scale factor is closer to

than to

, every point in the image will be closer to our center

than to its corresponding point in the pre-image.
Hint #44 / 4
Connecting the vertices of the image, we get the following mapping:
Triangle A B C. Point B is six units to the right of point P and three units down, Point C is three units left from point P and up six units, and Point A is six units left of point P. There is another triangle graphed that is triangle A B C with a scale factor of one third from point P. The new triangle the Point B prime is two units to the right of point P and one unit down, Point C prime is one unit left from point P and up two units, and Point A prime is two units left of point P. The distance between C prime and B prime is approximately four point two units. The distance between C prime and A prime is approximately two point two units. The distance between A prime and B prime is four point one units.







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22 of 30

under a dilation whose center is

and scale factor is


.

1 answer

It seems like you want to perform a dilation on a triangle \( ABC \) with respect to a point \( P \) as the center and a specific scale factor. While I cannot draw directly, I can help you understand how to calculate the coordinates of the image vertices after dilation.

Let's consider the information provided step by step:

  1. Identifying the coordinates of points:

    • Start by assigning coordinates to point \( P \). For convenience, let's say \( P = (0, 0) \).
    • Point \( B \) is given as \( 6 \) units right and \( 3 \) units down from \( P \). Therefore, the coordinates of Point \( B \) will be \( B = (6, -3) \).
    • Point \( C \) is \( 3 \) units left and \( 6 \) units up from \( P \). So, the coordinates of Point \( C \) will be \( C = (-3, 6) \).
    • Point \( A \) is \( 6 \) units left from \( P \), leading to coordinates \( A = (-6, 0) \).

  2. Dilation with respect to point \( P \):

    • You mentioned a scale factor (although it is not specified in your message). For this example, let's assume a scale factor of \( k \).
    • To find the coordinates of the dilated points:
      • The formula to find the new coordinates \( (x', y') \) after dilation is: \[ x' = P_x + k \cdot (x - P_x) \] \[ y' = P_y + k \cdot (y - P_y) \]
    • For each vertex:
      • For \( B(6, -3) \): \[ B' = (0 + k(6 - 0), 0 + k(-3 - 0)) = (6k, -3k) \]
      • For \( C(-3, 6) \): \[ C' = (0 + k(-3 - 0), 0 + k(6 - 0)) = (-3k, 6k) \]
      • For \( A(-6, 0) \): \[ A' = (0 + k(-6 - 0), 0 + k(0 - 0)) = (-6k, 0) \]

  3. Connecting the vertices:

    • After calculating these new coordinates, connect the points \( A', B', C' \) to form the new triangle after dilation.

  4. Finding distances:

    • To compute the distances between the new points \( A', B', C' \), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
    • For example, the distance \( BC \) can be calculated using the coordinates of \( B' \) and \( C' \).

This process will yield the image coordinates of the triangle after scaling and help you visualize or draw the transformation. If you have a specific scale factor or any additional details, feel free to share those for a more tailored response!