Given the following directional notations:

- $A \# (12) B$ means $A$ is $21 ext{ m}$ east of $B$
- $A \ast (03) B$ means $A$ is $30 ext{ m}$ north of $B$
- $A \% (24) B$ means $A$ is $42 ext{ m}$ west of $B$
- $A \@ (52) B$ means $A$ is $25 ext{ m}$ south of $B$

And the conditions:

- $X \# (93) Y$
- $T \ast (42) U$
- $R \@ (61) Q$
- $T \# (03) S$
- $P \% (14) Q$
- $W \@ (61) X$
- $Z \ast (01) Y$
- $W \% (45) U$
- $S \ast (80) R$

What is the shortest distance and direction of $S$ with respect to $X$?

Question

Given the following directional notations:

- $A \# (12) B$ means $A$ is $21 	ext{ m}$ east of $B$
- $A \ast (03) B$ means $A$ is $30 	ext{ m}$ north of $B$
- $A \% (24) B$ means $A$ is $42 	ext{ m}$ west of $B$
- $A \@ (52) B$ means $A$ is $25 	ext{ m}$ south of $B$

And the conditions:

- $X \# (93) Y$
- $T \ast (42) U$
- $R \@ (61) Q$
- $T \# (03) S$
- $P \% (14) Q$
- $W \@ (61) X$
- $Z \ast (01) Y$
- $W \% (45) U$
- $S \ast (80) R$

What is the shortest distance and direction of $S$ with respect to $X$?

Accepted Answer

**Solution:**

1. First, decode each directional statement into relative positions:

- $X \# (93) Y$ means $X$ is 21 m east of $Y$.
- $W \@ (61) X$ means $W$ is 25 m south of $X$.
- $T \# (03) S$ means $T$ is 21 m east of $S$.
- $S \ast (80) R$ means $S$ is 30 m north of $R$.
- $R \@ (61) Q$ means $R$ is 25 m south of $Q$.
- $P \% (14) Q$ means $P$ is 42 m west of $Q$.
- $W \% (45) U$ means $W$ is 42 m west of $U$.
- $T \ast (42) U$ means $T$ is 30 m north of $U$.
- $Z \ast (01) Y$ means $Z$ is 30 m north of $Y$.

2. Using these, place points stepwise on a coordinate plane:

- Let’s place $Y$ at origin $(0,0)$.
- Then $X$ is 21 m east of $Y$: $X(21,0)$.
- $Z$ is 30 m north of $Y$: $Z(0,30)$.
- $W$ is 25 m south of $X$: $W(21,-25)$.
- $W$ is 42 m west of $U$: so $U$ is 42 m east of $W$: $U(63,-25)$.
- $T$ is 30 m north of $U$: $T(63,5)$.
- $T$ is 21 m east of $S$: so $S$ is 21 m west of $T$: $S(42,5)$.
- $S$ is 30 m north of $R$: so $R$ is 30 m south of $S$: $R(42,-25)$.
- $R$ is 25 m south of $Q$: so $Q$ is 25 m north of $R$: $Q(42,0)$.
- $P$ is 42 m west of $Q$: $P(0,0)$.

3. Now, find the shortest distance and direction of $S(42,5)$ from $X(21,0)$:

- Distance:
  $$\sqrt{(42 - 21)^2 + (5 - 0)^2} = \sqrt{21^2 + 5^2} = \sqrt{441 + 25} = \sqrt{466} \approx 21.6 	ext{ m}$$

- Direction:
  - The change in x is positive (eastward).
  - The change in y is positive (northward).
  - So direction is north-east.

4. Among the options, none exactly matches approximately 22 m north-east.

**Answer:** None of these

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