Two triangles on a coordinate plane, demonstrating a transformation from triangle ABC to triangle A'B'C'.

Let’s look at the two triangles on the grid: triangle ABC and triangle A'B'C'.

We are going to check if triangle A’B’C’ is a scaled-up version (a dilation) of triangle ABC. To do that, we’ll compare the lengths of corresponding sides.

First, let’s find the side lengths of triangle ABC:

- Side AB: from A(1,1) to B(1,3). Since x-coordinates are the same, it’s vertical. Length = 3 - 1 = 2 units.
- Side AC: from A(1,1) to C(4,1). Since y-coordinates are the same, it’s horizontal. Length = 4 - 1 = 3 units.
- Side BC: from B(1,3) to C(4,1). We can use the distance formula or count squares. Let’s use the Pythagorean theorem:
Horizontal change: 4 - 1 = 3
Vertical change: 3 - 1 = 2
So length = √(3² + 2²) = √(9 + 4) = √13 ≈ 3.606 — but we don’t need the decimal yet.

Now, let’s find the side lengths of triangle A’B’C’:

- Side A’B’: from A’(3,3) to B’(3,9). Vertical line. Length = 9 - 3 = 6 units.
- Side A’C’: from A’(3,3) to C’(12,3). Horizontal line. Length = 12 - 3 = 9 units.
- Side B’C’: from B’(3,9) to C’(12,3).
Horizontal change: 12 - 3 = 9
Vertical change: 9 - 3 = 6
Length = √(9² + 6²) = √(81 + 36) = √117 = √(9×13) = 3√13

Now let’s compare corresponding sides:

- AB = 2 → A’B’ = 6 → ratio = 6 ÷ 2 = 3
- AC = 3 → A’C’ = 9 → ratio = 9 ÷ 3 = 3
- BC = √13 → B’C’ = 3√13 → ratio = 3√13 ÷ √13 = 3

All three sides are multiplied by the same scale factor: 3

Also, notice that point A moved from (1,1) to (3,3), which is adding 2 to both x and y. But since all sides scaled by 3, this is likely a dilation centered at some point — but the question doesn’t ask for center, just whether it’s a dilation with scale factor 3.

Since all corresponding sides are exactly 3 times longer, yes — triangle A’B’C’ is a dilation of triangle ABC with scale factor 3.

Final Answer: Yes, triangle A'B'C' is a dilation of triangle ABC with a scale factor of 3.