11. Write a sequence of transformations that maps triangle ABC onto triangle A''B''C''.

Here is ABC: A(1,8), B(3,12), C(4,4)

Here is A''B''C'': A''(3,-3), B''(5,-6) C''(6,2)

Please help I have tried everything and this is over due.

There is no choicies bc i have to write them in on the test. Please help me.

6 answers

jezuz people

Please i need this done.

V-School? i have the same question

Yep

ME 2

To map triangle ABC onto triangle A''B''C'', you can use a combination of translation, rotation, and reflection.

Step 1: Translation
To translate the triangle, you need to move every point by the same amount in both the x and y directions. Determine the translation vector by subtracting the corresponding coordinates of the corresponding points:

Vector AB: (3,12) - (1,8) = (2,4)
Vector AC: (4,4) - (1,8) = (3, -4)

Using these translation vectors, you can translate triangle ABC to a new position:

A'(1,8) + (2,4) = (3, 12)
B'(3,12) + (2,4) = (5, 16)
C'(4,4) + (3,-4) = (7, 0)

Step 2: Rotation
To rotate the triangle, you need to determine the angle and pivot point around which the rotation occurs. Let's assume the rotation angle is 180 degrees and the pivot point is the centroid of triangle ABC. To find the centroid, calculate the average of the x-coordinates and the average of the y-coordinates:

Centroid: ((1+3+4)/3, (8+12+4)/3) = (8/3, 24/3) = (8/3, 8)

Next, rotate each translated point around the centroid by 180 degrees:

A'': (8/3, 8) + 180 degrees rotation = (8/3, 8) - (3,8) = (8/3 - 3, 8 - 8) = (-1/3, 0)
B'': (5, 16) - (3,8) = (5-3, 16-8) = (2, 8)
C'': (7, 0) - (3,8) = (7-3, 0-8) = (4, -8)

Step 3: Reflection
Lastly, determine the reflection of the rotated triangle over the x-axis or y-axis to match the coordinates of triangle A''B''C''. In this case, we reflect it over the x-axis.

Reflect each rotated point over the x-axis:

A'''': (-1/3, 0) reflect over x-axis = (-1/3, -0) = (-1/3, 0)
B'''': (2, 8) reflect over x-axis = (2, -8)
C'''': (4, -8) reflect over x-axis = (4, 8)

So, the sequence of transformations that maps triangle ABC onto triangle A''B''C'' is:
1. Translation: (2, 4) in the positive x and y directions
2. Rotation: 180 degrees counterclockwise around the centroid (8/3, 8)
3. Reflection: Reflection over the x-axis

Applying these transformations to triangle ABC would result in triangle A''B''C''.