Describe the single transformation represented by the composition of a reflection on the line y = -x and a rotation about the origin through 180 degrees. Then, find the image of triangle ABC with vert

Here's the solution to the transformation problem:

First, let's define the individual transformations:

• R (Reflection on the line $y = -x$): A point $(x, y)$ is transformed to $(-y, -x)$.
• S (Rotation about the origin through $180^\circ$): A point $(x, y)$ is transformed to $(-x, -y)$.

The combined transformation is RoS, which means we apply S first, then R.

Part 1: Find the single transformation represented by RoS.

Let's apply the transformations to a general point $(x, y)$:

1. Apply S to $(x, y)$: $$S(x, y) = (-x, -y)$$
2. Now, apply R to the result $(-x, -y)$. Let $x' = -x$ and $y' = -y$. Using the rule for R, which is $(x', y') \to (-y', -x')$, we get: $$R(-x, -y) = (-(-y), -(-x)) = (y, x)$$

So, the combined transformation RoS maps a point $(x, y)$ to $(y, x)$.
This single transformation is a reflection on the line $y = x$.

The single transformation represented by RoS is $\boxed{\text{Reflection on the line } y=x}$.

Part 2: Find the co-ordinates of the vertices of the image of $\triangle ABC$.

The vertices of $\triangle ABC$ are A(1, 2), B(-4, 3), and C(3, 5).
We use the single transformation rule $RoS(x, y) = (y, x)$ to find the image coordinates.

• For vertex A(1, 2): $$A' = RoS(1, 2) = (2, 1)$$
• For vertex B(-4, 3): $$B' = RoS(-4, 3) = (3, -4)$$
• For vertex C(3, 5): $$C' = RoS(3, 5) = (5, 3)$$

The coordinates of the image $\triangle A'B'C'$ are $\boxed{A'(2, 1), B'(3, -4), C'(5, 3)}$.

Part 3: Present $\triangle ABC$ and its image on the same graph paper.

To present this on graph paper, you would plot the following points and connect them to form the triangles:

• Original Triangle $\triangle ABC$:
  • A(1, 2)
  • B(-4, 3)
  • C(3, 5)
• Image Triangle $\triangle A'B'C'$:
  • A'(2, 1)
  • B'(3, -4)
  • C'(5, 3)

Draw the line $y=x$ on your graph paper. You will observe that $\triangle A'B'C'$ is the reflection of $\triangle ABC$ across the line $y=x$.