TRANSFORMATIONS QUESTION HELP!!

Question

Jade is working on a puzzle for a company who makes games. She is working on a program that uses transformations to move the puzzle shown by TRIANGLE ABC into the target area shown by TRIANGLE DEF.

 

A(-5,4)

B(-3,4)

C(-3,1)

 

D(4,-3)

E(4,-1)

F(1,-1)

 

Find a sequence of transformations that moves the puzzle piece to the target area for Case 1 and Case 2. USE AS FEW MOVES AS POSSIBLE. 

 

Case 1: Reflections are allowed only across vertical and horizontal lines.

 

Case 2: Reflections across vertical and horizontal lines are not allowed. 

 

PLEASE HELP! THANK YOU 

 

Answer 1

Both cases can be solved using exactly 2 transformations, which is the minimum number of moves.

Case 1:

Minimum: 2 moves

Move 1: Rotate triangle ABC 90° counterclockwise about the point \((0,2)\).

The new coordinates are:

1. A′ = (−2, −3)
2. B′ = (−2, −1)
3. C′ = (1, −1)

Move 2: Reflect the resulting triangle across the vertical line x = 1.

The final coordinates are:

1. A″ = (4, −3) = D
2. B″ = (4, −1) = E
3. C″ = (1, −1) = F

This matches triangle DEF exactly.

Case 2:

Minimum: 2 moves

Move 1: Reflect triangle ABC across the diagonal line y = x.

The new coordinates are:

1. A′ = (4, −5)
2. B′ = (4, −3)
3. C′ = (1, −3)

Move 2: Translate the resulting triangle 2 units upward.

The final coordinates are:

1. A″ = (4, −3) = D
2. B″ = (4, −1) = E
3. C″ = (1, −1) = F

This also matches triangle DEF exactly.

So in short the answer is:

Case 1:

First transformation: Rotate 90° counterclockwise about (0, 2)

Second transformation: Reflect across x = 1

Case 2:

First transformation: Reflect across y = x

Second transformation: Translate 2 units up