Create a sequence of transformations that can map a triangle PQR onto a triangle P'Q'R' shown in the picture below by performing rotations and dilations.
To map triangle PQR onto triangle P'Q'R', we can use a sequence of rotations and dilations. Here is one possible sequence of transformations:
1. Translation: Move point P to the origin (0, 0) by subtracting the coordinates of P from all points. This ensures that the origin becomes the new P.
P' = P - P = (0, 0)
Q' = Q - P
R' = R - P
2. Rotation: Rotate triangle P'Q'R' such that line P'Q' overlaps with line P'Q. This rotation will align the two triangles along Q'
Find the angle between the line P'Q' and the x-axis.
Rotate points Q' and R' by this angle.
Let theta be the angle between the x-axis and the line P'Q'
Q'' = (Q'.x * cos(theta) - Q'.y * sin(theta), Q'.x * sin(theta) + Q'.y * cos(theta))
R'' = (R'.x * cos(theta) - R'.y * sin(theta), R'.x * sin(theta) + R'.y * cos(theta))
3. Scaling (Dilation): Scale triangle P''Q''R'' using a dilation centered at the origin. This will make the lengths of P''Q'' and Q''R'' match the lengths of P'Q' and QR respectively.
Determine the scaling factor to make P''Q'' have the same length as P'Q'.
Let scale = length(P'Q') / length(P''Q'')
P''' = (P''.x * scale, P''.y * scale)
Q''' = (Q''.x * scale, Q''.y * scale)
R''' = (R''.x * scale, R''.y * scale)
4. Translation: Move the origin back to its original position by adding the coordinates of P to all points. This will restore the triangle to its original location.
P'''' = P''' + P
Q'''' = Q''' + P
R'''' = R''' + P
Now, triangle PQR has been mapped onto triangle P'Q'R'.
Note: The exact calculations and angles may vary depending on the coordinates and dimensions of the given triangles.
To map a triangle PQR onto a triangle P'Q'R' using rotations and dilations, you can follow these steps:
Step 1: Translation
If the two triangles are not already oriented in the same direction, you can start by translating triangle PQR to align one of the vertices (let's say vertex P) with the corresponding vertex P' on the other triangle.
Step 2: Rotation
After the translation, you may need to rotate the triangle to match the orientation of the other triangle. To do this, choose a point O as the center of rotation and determine the angle of rotation. Let's say you want to rotate around O by θ degrees.
To find the angle θ, you can use the coordinates of the two vertices you want to align (for example, Q and Q') using the following formula:
θ = arctan((yQ' - yO)/(xQ' - xO)) - arctan((yQ - yO)/(xQ - xO))
This formula calculates the difference in angles between the two vertices. Once you find the angle θ, you can rotate the entire triangle PQR by θ degrees clockwise or counterclockwise around point O.
Step 3: Dilation
The final step is to perform a dilation to resize triangle PQR to match the size of triangle P'Q'R'. A dilation involves scaling the triangle by a certain factor. This factor can be determined by comparing the lengths of corresponding sides on both triangles.
For example, if you want to dilate triangle PQR to make side PQ match side P'Q', you can calculate the scale factor as:
Scale factor = length(P'Q') / length(PQ)
Once you have the scale factor, you can apply it to all the sides of triangle PQR to resize it accordingly.
By following these steps of translation, rotation, and dilation, you can map triangle PQR onto triangle P'Q'R' using rotations and dilations.