1. Perform these transformations on ∆TRL and compute the perimeter of the pre-image and final image. Show your computation for finding the image vertices and for finding the perimeters.
a. Rotate ∆TRL 180¢X about the origin and label the resulting image ∆T¡¦ R¡¦ L¡¦.
b. Reflect ∆T¡¦ R¡¦ L¡¦ (the image from a) across the y-axis and label the resulting image ∆T¡¦¡¦ R¡¦¡¦ L¡¦¡¦.
c. Translate ∆T¡¦¡¦ R¡¦¡¦ L¡¦¡¦ (the image from b) according to (x, y) „_(x + 3, y ¡V 3) and label the image ∆T¡¦¡¦¡¦ R¡¦¡¦¡¦ L¡¦¡¦¡¦.
To perform the transformations on triangle ∆TRL and compute the perimeter of the pre-image and final image, we need to follow these steps:
Step 1: Determine the coordinates of the vertices of triangle ∆TRL.
Let's say the coordinates of point T are (x1, y1), point R are (x2, y2), and point L are (x3, y3).
Step 2: Rotate ∆TRL 180° about the origin (0, 0).
To rotate a point (x, y) 180° about the origin, we simply change the signs of both the x and y coordinates. So, the new coordinates for the rotated image ∆T' R' L' are:
T' = (-x1, -y1)
R' = (-x2, -y2)
L' = (-x3, -y3)
Step 3: Calculate the perimeter of the pre-image ∆TRL.
The perimeter of a triangle is the sum of the lengths of its sides. We can use the distance formula to find the lengths of the sides and then add them up.
Let's calculate the lengths of the sides:
Side TR = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Side RL = sqrt((x3 - x2)^2 + (y3 - y2)^2)
Side LT = sqrt((x1 - x3)^2 + (y1 - y3)^2)
The perimeter of pre-image ∆TRL = Side TR + Side RL + Side LT
Step 4: Calculate the perimeter of the final image ∆T'' R'' L''.
We will continue with the transformed image ∆T' R' L' obtained from step 2.
Reflecting across the y-axis simply means changing the sign of the x-coordinate for each point:
T'' = (-x1, y1)
R'' = (-x2, y2)
L'' = (-x3, y3)
Now calculate the lengths of the sides using the distance formula as we did in step 3.
The perimeter of final image ∆T'' R'' L'' = Side T''R'' + Side R''L'' + Side L''T''
Step 5: Translate ∆T'' R'' L'' by (x, y) ➝ (x + 3, y - 3).
To translate a point (x, y) by (a, b), we add a to the x-coordinate and b to the y-coordinate. So, the new coordinates for the translated image ∆T''' R''' L''' are:
T''' = (x1 + 3, y1 - 3)
R''' = (x2 + 3, y2 - 3)
L''' = (x3 + 3, y3 - 3)
Calculate the lengths of the sides for ∆T''' R''' L''' as we did in step 3.
The perimeter of the final translated image ∆T''' R''' L''' = Side T'''R''' + Side R'''L''' + Side L'''T'''
By following these steps, you can compute the perimeters of the pre-image and final image for each transformation.