Let's analyze and solve the problems step-by-step:

Given Information:

1. Height of the person: 1.5 meters.
2. Route:
   • P1 to P2 (horizontal walk).
   • P2 to P3 (uphill walk).

Problem Breakdown:

(1) Determine the actual coordinates of P2.

Given that $x$ represents the horizontal distance from L, and $y$ represents the shadow length in meters.

• From P1 to P2, it is a horizontal walk, hence $y = a$.
• At P2, we have coordinates $(6, a)$.

(2) Find the height of the lamp $L$.

• From the first graph, the height of the lamp $L$ can be calculated using the similar triangles principle.
• The height of the shadow at $P2$ is equal to $a$.

(3) Given $P2P3$ forms an angle $\alpha$ with the horizontal, find $\alpha$ when it is $\theta$.

• The angle of $45^\circ$ forms with the horizontal distance and height.

(4) Calculate the slope of $OA$ and $BC$.

• Slope of $OA$: Using the coordinates of $A(6, a)$, the slope will be $\frac{a - 0}{6 - 0} = \frac{a}{6}$.
• Slope of $BC$: Using the coordinates of $B(8, b)$ and $C(14, c)$, find the difference in coordinates.

Solutions:

1. Coordinates of $P2$: The actual coordinates can be interpreted as $P2(6, a)$.

2. Height of Lamp $L$: Using similar triangles for shadow: $\frac{h}{d} = \frac{1.5}{y}$ where $h$ is the height of the lamp and $d$ is the distance from $P1$ to the person.

3. Angle Calculation: If $\alpha = 45^\circ$, then: $\tan(45^\circ) = \frac{\text{height change}}{\text{horizontal distance}}$ and use the given distances to calculate.

4. Slope Calculation: $\text{Slope of } OA = \frac{a}{6}$ $\text{Slope of } BC = \frac{c - b}{14 - 8}$

Let's compute the height of the lamp, the coordinates, and slopes.

Detailed Steps:

1. Coordinates of $P2$:

Given coordinates $P2(6, a)$.

2. Height of Lamp $L$:

Using the shadow principle: $\frac{L}{6} = \frac{1.5}{a} \implies L = 6 \times \frac{1.5}{a}$

3. Angle Calculation:

For $\alpha = 45^\circ$, $\tan(45^\circ) = 1$: $1 = \frac{\Delta y}{\Delta x}$

4. Slopes:

Slope of $OA$: $\frac{a}{6}$

Slope of $BC$: $\frac{c - b}{14 - 8} = \frac{c - b}{6}$

If we assume $a = 3$, $b = 4$, and $c = 7$, the calculations will be specific.

Would you like to see the calculations for specific values or have any questions? Here are five questions for further inquiry:

1. Can you provide the specific values of $a$, $b$, and $c$ for precise calculation?
2. How should we interpret the distances for the shadow and height in different scenarios?
3. Do you need a step-by-step calculation for finding the height of the lamp?
4. Would you like to know more about how to derive the slope using coordinates?
5. Do you need further clarification on similar triangles and shadow calculations?

Tip:

When solving geometry problems involving shadows and heights, always look for similar triangles to simplify the calculations.