Assertion (A): The length of the line AB is 100 m.

Reason (R): tan 45° = $\dfrac{50}{PB}$ ⇒ PB = 50 m and AB = 50 m + 50 m = 100 m

1. A is true, but R is false.

2. A is false, but R is true.

3. Both A and R are true, and R is the correct reason for A.

4. Both A and R are true, and R is the incorrect reason for A.

From figure,

⇒ BC = AP

⇒ AP = 50 m

By formula,

tan θ = $\dfrac{\text{Perpendicular}}{\text{Base}}$

In ΔPBC,

$\Rightarrow \text{tan 45°} = \dfrac{BC}{PB}\\[1em] \Rightarrow 1 = \dfrac{50}{PB}\\[1em] \Rightarrow PB = 50 \text{ m}.$

From figure,

AB = AP + PB = 50 m + 50 m = 100 m.

∴ Both A and R are true, and R is the correct reason for A.

Hence, option 3 is the correct option.