Triangle PQR has vertices P (-2, 5), Q(1, 3) and R (0, -1). (a) Triangle P'Q'R' is the image of PQR after a reflection in the line y=x. On the grid, draw triangles PQR and P'Q'R'.

(a) Triangle PQR has vertices $P(-2, 5)$, $Q(1, 3)$, and $R(0, -1)$.

To find the image $P'Q'R'$ after a reflection in the line $y = x$, swap the coordinates of each point.

• $P' = (5, -2)$
• $Q' = (3, 1)$
• $R' = (-1, 0)$

(b) Triangle $P''Q''R''$ is the image of $P'Q'R'$ after a rotation of $180^\circ$ about the point $(2, 1)$.

To rotate $180^\circ$, use the formula:

$$(x', y') = (2 - (x - 2), 1 - (y - 1))$$

• $P'' = (2 - (5 - 2), 1 - (-2 - 1)) = (-1, 4)$
• $Q'' = (2 - (3 - 2), 1 - (1 - 1)) = (1, 1)$
• $R'' = (2 - (-1 - 2), 1 - (0 - 1)) = (5, 2)$

(c) Triangle $P'''Q'''R'''$ is the image of $P''Q''R''$ after an enlargement, scale factor $\frac{1}{2}$, center $(-4, 4)$.

Use the enlargement formula:

$$(x', y') = \left(-4 + \frac{1}{2}(x + 4), 4 + \frac{1}{2}(y - 4)\right)$$

• $P''' = \left(-4 + \frac{1}{2}(-1 + 4), 4 + \frac{1}{2}(4 - 4)\right) = (-2.5, 4)$
• $Q''' = \left(-4 + \frac{1}{2}(1 + 4), 4 + \frac{1}{2}(1 - 4)\right) = (-1.5, 2.5)$
• $R''' = \left(-4 + \frac{1}{2}(5 + 4), 4 + \frac{1}{2}(2 - 4)\right) = (0.5, 3)$

(d) The single transformation that maps $PQR$ directly onto $P'''Q'''R'''$ is a combination of transformations. However, if considering a single transformation, it could be described as a reflection followed by a rotation and enlargement, but this is not a single transformation.