Sequence of Transformations | Order, Identification & Examples ... - Free Printable

Let’s carefully analyze the problem.

We are given three triangles on a coordinate plane:

- Triangle ABC (blue) — original
- Triangle A'B'C' (orange) — transformed version of ABC
- Triangle A''B''C'' (green) — another transformed version

We need to figure out what transformations were applied to get from triangle ABC to A’B’C’, and then to A’’B’’C’’. But since no explicit question is written in the image, we must infer the most likely task: identify the transformations that map triangle ABC → A’B’C’ → A’’B’’C’’.

Let’s find coordinates first.

From the grid:

Triangle ABC:
- A = (-3, 1)
- B = (-4, 5)
- C = (-1, 3)

Triangle A’B’C’:
- A’ = (3, 1)
- B’ = (4, 5)
- C’ = (1, 3)

Compare ABC to A’B’C’:

Look at point A(-3,1) → A’(3,1): x changed sign, y same → reflection over y-axis?
Check B(-4,5) → B’(4,5): yes, x negated, y same → reflection over y-axis.
C(-1,3) → C’(1,3): same pattern.

✔ So, transformation from ABC to A’B’C’ is reflection over the y-axis.

Now, triangle A’’B’’C’’:
- A’’ = (0, -4)
- B’’ = (1, 0)
- C’’ = (-2, -2)

Wait — this doesn’t look like a simple reflection or translation of A’B’C’. Let’s check if it’s a rotation or something else.

But hold on — maybe A’’B’’C’’ is a transformation of the *original* ABC? Or perhaps of A’B’C’?

Let’s try mapping ABC directly to A’’B’’C’’.

A(-3,1) → A’’(0,-4)
B(-4,5) → B’’(1,0)
C(-1,3) → C’’(-2,-2)

Let’s compute vectors:

From A to A’’: (0 - (-3), -4 - 1) = (3, -5)
From B to B’’: (1 - (-4), 0 - 5) = (5, -5) ← not same vector → not translation.

Try rotation? Maybe around origin?

Rotate A(-3,1) 90° clockwise: (x,y) → (y, -x) → (1, 3) → not A’’(0,-4)

Rotate 90° counterclockwise: (x,y) → (-y, x) → (-1, -3) → nope.

What about reflecting over x-axis then translating?

Reflect ABC over x-axis:
A(-3,1) → (-3,-1)
B(-4,5) → (-4,-5)
C(-1,3) → (-1,-3)

Then translate by (3, -3)?
(-3+3, -1-3) = (0,-4) → matches A’’
(-4+3, -5-3) = (-1,-8) → but B’’ is (1,0) → no.

Alternative idea: Maybe A’’B’’C’’ is obtained by rotating ABC 180° around some point?

Or perhaps it's a different transformation altogether.

Wait — let’s look again at A’B’C’ to A’’B’’C’’.

A’(3,1) → A’’(0,-4)
B’(4,5) → B’’(1,0)
C’(1,3) → C’’(-2,-2)

Compute differences:

A’ to A’’: (0-3, -4-1) = (-3, -5)
B’ to B’’: (1-4, 0-5) = (-3, -5)
C’ to C’’: (-2-1, -2-3) = (-3, -5)

✔ All points moved by vector (-3, -5)!

So transformation from A’B’C’ to A’’B’’C’’ is translation by vector <-3, -5>.

Therefore, overall:

ABC → (reflect over y-axis) → A’B’C’ → (translate by <-3, -5>) → A’’B’’C’’

Alternatively, we could describe the full transformation from ABC to A’’B’’C’’ as:

First reflect over y-axis, then translate left 3 and down 5.

But since the problem likely wants us to identify each step, and based on standard curriculum tasks, this is probably the expected answer.

Let me verify one more time with actual numbers.

Take point A(-3,1):

Step 1: Reflect over y-axis → (3,1) → which is A’ ✔️
Step 2: Translate by <-3, -5> → (3-3, 1-5) = (0,-4) → which is A’’ ✔️

Point B(-4,5):

Reflect → (4,5) → B’ ✔️
Translate → (4-3, 5-5) = (1,0) → B’’ ✔️

Point C(-1,3):

Reflect → (1,3) → C’ ✔️
Translate → (1-3, 3-5) = (-2,-2) → C’’ ✔️

Perfect.

So the two transformations are:

1. Reflection over the y-axis (to get from ABC to A’B’C’)
2. Translation by vector <-3, -5> (to get from A’B’C’ to A’’B’’C’’)

Since the user didn't specify exactly what to find, but given the context of such diagrams in school math, the most reasonable interpretation is to describe the sequence of transformations.

Final Answer:

The triangle ABC was first reflected over the y-axis to form triangle A'B'C', and then translated 3 units left and 5 units down to form triangle A''B''C''.