Patterns and Algebra - Grid References and Compass Directions

A treasure chest is located at $(3, 5)$ on a map. If a pirate starts at $(1, 2)$, how many units East and how many units North must they travel to reach the chest?

$2 \text{ units East, } 3 \text{ units North}$

To find the distance, subtract the starting coordinates from the target coordinates. For the $x$-direction (East): $3 - 1 = 2$. For the $y$-direction (North): $5 - 2 = 3$.

A robot follows the algebraic rule $(x, y) \rightarrow (x + 4, y - 2)$. If the robot starts at position $P(2, 6)$, find its new position $P'$.

$P'(6, 4)$

Substitute the initial coordinates into the rule: $x_{new} = 2 + 4 = 6$ and $y_{new} = 6 - 2 = 4$. The new point is $(6, 4)$.

An explorer is facing North. They turn $225^\circ$ clockwise. In which direction are they now facing?

$SW \text{ (Southwest)}$

A $180^\circ$ turn clockwise from North faces South. An additional $45^\circ$ turn ($225^\circ - 180^\circ = 45^\circ$) from South towards the West leads to the Southwest ($SW$) direction.

Starting at $(5, 8)$, a point moves 3 units West and 4 units South. What are the final coordinates?

$(2, 4)$

Moving West decreases the $x$-coordinate: $5 - 3 = 2$. Moving South decreases the $y$-coordinate: $8 - 4 = 4$. The final position is $(2, 4)$.