To solve this question, let's go through each part step-by-step.

Part (a)

We are asked to draw triangle $PQR$ on the grid with given vertices:

• $P(5, -1)$
• $Q(3, -1)$
• $R(3, -3)$

1. Locate each of these points on the grid and plot them.
2. Connect the points $P$, $Q$, and $R$ to form the triangle $PQR$.

Part (b)

In this part, we need to draw $\triangle P'Q'R'$, which is the image of $\triangle PQR$ after a reflection in the line $y = x$.

To reflect points over the line $y = x$:

1. Swap the $x$- and $y$-coordinates of each point.

Thus, the reflected coordinates for each vertex are:

• $P(5, -1)$ becomes $P'(-1, 5)$
• $Q(3, -1)$ becomes $Q'(-1, 3)$
• $R(3, -3)$ becomes $R'(-3, 3)$

Now plot these points $P'(-1, 5)$, $Q'(-1, 3)$, and $R'(-3, 3)$ on the grid and connect them to form $\triangle P'Q'R'$.

Part (c)

In this part, we are given a transformation matrix $T$: $T = \begin{pmatrix} 1.5 & -0.5 \\ -0.5 & 1.5 \end{pmatrix}$ We are to apply this transformation to the vertices of $\triangle P'Q'R'$ to obtain the coordinates of $\triangle P''Q''R''$.

To find the transformed coordinates, we multiply each vertex coordinate of $\triangle P'Q'R'$ by the matrix $T$.

1. For $P'(-1, 5)$: $\begin{pmatrix} 1.5 & -0.5 \\ -0.5 & 1.5 \end{pmatrix} \begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} (1.5 \times -1) + (-0.5 \times 5) \\ (-0.5 \times -1) + (1.5 \times 5) \end{pmatrix} = \begin{pmatrix} -1.5 - 2.5 \\ 0.5 + 7.5 \end{pmatrix} = \begin{pmatrix} -4 \\ 8 \end{pmatrix}$ So, $P''(-4, 8)$.

2. For $Q'(-1, 3)$: $\begin{pmatrix} 1.5 & -0.5 \\ -0.5 & 1.5 \end{pmatrix} \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} (1.5 \times -1) + (-0.5 \times 3) \\ (-0.5 \times -1) + (1.5 \times 3) \end{pmatrix} = \begin{pmatrix} -1.5 - 1.5 \\ 0.5 + 4.5 \end{pmatrix} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$ So, $Q''(-3, 5)$.

3. For $R'(-3, 3)$: $\begin{pmatrix} 1.5 & -0.5 \\ -0.5 & 1.5 \end{pmatrix} \begin{pmatrix} -3 \\ 3 \end{pmatrix} = \begin{pmatrix} (1.5 \times -3) + (-0.5 \times 3) \\ (-0.5 \times -3) + (1.5 \times 3) \end{pmatrix} = \begin{pmatrix} -4.5 - 1.5 \\ 1.5 + 4.5 \end{pmatrix} = \begin{pmatrix} -6 \\ 6 \end{pmatrix}$ So, $R''(-6, 6)$.

Final Coordinates for $\triangle P''Q''R''$

The vertices of $\triangle P''Q''R''$ are:

• $P''(-4, 8)$
• $Q''(-3, 5)$
• $R''(-6, 6)$

Would you like a detailed explanation of each step or further assistance on plotting these points?

Here are some related questions for further exploration:

1. What would be the coordinates of $\triangle PQR$ if it were reflected over the line $y = -x$ instead?
2. How would the transformation matrix change if we wanted to apply a rotation instead of a shear?
3. How does reflecting over $y = x$ compare to reflecting over the $x$-axis in terms of coordinate transformation?
4. If we apply the transformation matrix $T$ twice, what would be the new coordinates of $P''Q''R''$?
5. How would scaling the transformation matrix $T$ by 2 affect the coordinates of $\triangle P''Q''R''$?

Tip: When applying transformations, always check if the operation is linear (like rotations and reflections) or affine, as this can impact the final coordinates significantly.