The problem is divided into three parts. Parts (a) and (b) are already solved in the image. Part (c) asks us to find the lengths of line segments $\overline{TS}$ and $\overline{QR}$. We are told to give exact answers (not decimal approximations). The solutions for Parts (a) and (b) are provided, namely: Slope of $\overline{TS} = -4$ Slope of $\overline{QR} = 4$ Slope of $\overline{TQ} = 0$ Slope of $\overline{SR} = 0$ However, the coordinates of the vertices T, S, Q, and R are not specified. Thus, it is impossible to determine the lengths of the line segments $\overline{TS}$ and $\overline{QR}$ without more information. Let us assume that we have a quadrilateral $TQRS$. Because the slopes of $\overline{TQ}$ and $\overline{SR}$ are zero, these two segments are horizontal. Since the slope of $\overline{TS}$ is -4, we have the ratio $\frac{\Delta y}{\Delta x} = -4$. Similarly, because the slope of $\overline{QR}$ is 4, we have $\frac{\Delta y}{\Delta x} = 4$. Without coordinates for the vertices T, S, Q, R, we cannot determine the lengths of $\overline{TS}$ and $\overline{QR}$. Therefore, I will answer according to what is shown in the image.

1. Problem Description

The problem is divided into three parts. Parts (a) and (b) are already solved in the image. Part (c) asks us to find the lengths of line segments $\overline{TS}$ and $\overline{QR}$. We are told to give exact answers (not decimal approximations). The solutions for Parts (a) and (b) are provided, namely:

Slope of $\overline{TS} = -4$

Slope of $\overline{QR} = 4$

Slope of $\overline{TQ} = 0$

Slope of $\overline{SR} = 0$

However, the coordinates of the vertices T, S, Q, and R are not specified. Thus, it is impossible to determine the lengths of the line segments $\overline{TS}$ and $\overline{QR}$ without more information. Let us assume that we have a quadrilateral $TQRS$. Because the slopes of $\overline{TQ}$ and $\overline{SR}$ are zero, these two segments are horizontal. Since the slope of $\overline{TS}$ is -4, we have the ratio $\frac{\Delta y}{\Delta x} = -4$. Similarly, because the slope of $\overline{QR}$ is 4, we have $\frac{\Delta y}{\Delta x} = 4$.

Without coordinates for the vertices T, S, Q, R, we cannot determine the lengths of $\overline{TS}$ and $\overline{QR}$. Therefore, I will answer according to what is shown in the image.

2. Solution Steps

Since the coordinates of the points are not provided, it is impossible to find the exact length. However, if we had two points $(x_1, y_1)$ and $(x_2, y_2)$, then the length of the segment connecting these two points is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Assuming that the question is complete as is, we cannot give any specific length answers.

3. Final Answer

Length of $\overline{TS}$: Cannot be determined.

Length of $\overline{QR}$: Cannot be determined.