Students plot points and calculate the length of line segments connecting them on a coordinate plane.

Problem Analysis:

The task involves solving problems related to plotting points in the coordinate plane, finding the length of line segments connecting two points, and calculating the perimeter of geometric figures given their vertices. Let's break this down step by step.

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#### Part 1: Plotting Points and Finding Line Segment Lengths

We are given pairs of points and asked to find the length of the line segment connecting them. The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

##### Problem 1:
Points: \( Q(3, 4) \) and \( R(3, 9) \)

- Coordinates of \( Q \): \( (3, 4) \)
- Coordinates of \( R \): \( (3, 9) \)

Using the distance formula:

\[
d = \sqrt{(3 - 3)^2 + (9 - 4)^2} = \sqrt{0^2 + 5^2} = \sqrt{25} = 5
\]

So, the length of the line segment \( QR \) is:

\[
\boxed{5}
\]

##### Problem 2:
Points: \( L(3, 3) \) and \( M(1, 8) \)

- Coordinates of \( L \): \( (3, 3) \)
- Coordinates of \( M \): \( (1, 8) \)

Using the distance formula:

\[
d = \sqrt{(1 - 3)^2 + (8 - 3)^2} = \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}
\]

So, the length of the line segment \( LM \) is:

\[
\boxed{\sqrt{29}}
\]

##### Problem 3:
Points: \( S(6, 0) \) and \( T(0, 6) \)

- Coordinates of \( S \): \( (6, 0) \)
- Coordinates of \( T \): \( (0, 6) \)

Using the distance formula:

\[
d = \sqrt{(0 - 6)^2 + (6 - 0)^2} = \sqrt{(-6)^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
\]

So, the length of the line segment \( ST \) is:

\[
\boxed{6\sqrt{2}}
\]

##### Problem 4:
Points: \( U(2, 2) \) and \( V(10, 10) \)

- Coordinates of \( U \): \( (2, 2) \)
- Coordinates of \( V \): \( (10, 10) \)

Using the distance formula:

\[
d = \sqrt{(10 - 2)^2 + (10 - 2)^2} = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}
\]

So, the length of the line segment \( UV \) is:

\[
\boxed{8\sqrt{2}}
\]

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#### Part 2: Finding the Perimeter of Figures

We are given the vertices of geometric figures and asked to find their perimeters. The perimeter is the sum of the lengths of all sides. We will use the distance formula to calculate the lengths of the sides.

##### Problem 5:
Vertices: \( R(2, 2) \), \( S(5, 2) \), \( T(5, 1) \), and \( U(2, 1) \)

This forms a rectangle. We calculate the lengths of the sides:

1. \( RS \):
\[
RS = \sqrt{(5 - 2)^2 + (2 - 2)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]

2. \( ST \):
\[
ST = \sqrt{(5 - 5)^2 + (1 - 2)^2} = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1
\]

3. \( TU \):
\[
TU = \sqrt{(2 - 5)^2 + (1 - 1)^2} = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3
\]

4. \( UR \):
\[
UR = \sqrt{(2 - 2)^2 + (1 - 2)^2} = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1
\]

The perimeter is the sum of these side lengths:

\[
\text{Perimeter} = RS + ST + TU + UR = 3 + 1 + 3 + 1 = 8
\]

So, the perimeter is:

\[
\boxed{8}
\]

##### Problem 6:
Vertices: \( J(1, 3) \), \( K(3, 5) \), \( L(5, 3) \), \( M(3, 1) \), and \( N(1, 3) \)

This forms a pentagon. We calculate the lengths of the sides:

1. \( JK \):
\[
JK = \sqrt{(3 - 1)^2 + (5 - 3)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

2. \( KL \):
\[
KL = \sqrt{(5 - 3)^2 + (3 - 5)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

3. \( LM \):
\[
LM = \sqrt{(3 - 5)^2 + (1 - 3)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

4. \( MN \):
\[
MN = \sqrt{(1 - 3)^2 + (3 - 1)^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

5. \( NJ \):
\[
NJ = \sqrt{(1 - 1)^2 + (3 - 3)^2} = \sqrt{0^2 + 0^2} = \sqrt{0} = 0
\]

However, since \( N \) and \( J \) are the same point, we do not include \( NJ \) in the perimeter calculation. The perimeter is the sum of the other four sides:

\[
\text{Perimeter} = JK + KL + LM + MN = 2\sqrt{2} + 2\sqrt{2} + 2\sqrt{2} + 2\sqrt{2} = 8\sqrt{2}
\]

So, the perimeter is:

\[
\boxed{8\sqrt{2}}
\]

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Final Answers:

1. \( \boxed{5} \)
2. \( \boxed{\sqrt{29}} \)
3. \( \boxed{6\sqrt{2}} \)
4. \( \boxed{8\sqrt{2}} \)
5. \( \boxed{8} \)
6. \( \boxed{8\sqrt{2}} \)