An ant walks inside a $18 \mathrm{~cm}$ by $150 \mathrm{~cm}$ rectangle. The ant's path follows straight lines which always make angles of $45^{\circ}$ to the sides of the rectangle. The ant starts from a point $X$ on one of the shorter sides. The first time the ant reaches the opposite side, it arrives at the midpoint. What is the distance, in centimetres, from $X$ to the nearest corner of the rectangle?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 9
An ant walks inside a $18 \mathrm{~cm}$ by $150 \mathrm{~cm}$ rectangle. The ant's path follows straight lines which always make angles of $45^{\circ}$ to the sides of the rectangle. The ant starts from a point $X$ on one of the shorter sides. The first time the ant reaches the opposite side, it arrives at the midpoint. What is the distance, in centimetres, from $X$ to the nearest corner of the rectangle?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 9

## Solution

If we took a movie of the ant's path and then played it backwards, the ant would now start at the point $E$ and would then end up at point $X$. Since the ant now 'starts' at a point nine $\mathrm{cm}$ from the corner, the 'first' part of his journey is from $E$ to $B$. This amounts to nine $\mathrm{cm}$ along the length of the rectangle since $\triangle B A E$ is an isosceles right-angled triangle. This process continues as illustrated, until the ant reaches point $C$. By the time the ant has reached $C$, it has travelled $9+18+3 \times 36$ or $135 \mathrm{~cm}$ along the length of the rectangle. To travel from $C$ to $X$, the ant must travel $15 \mathrm{~cm}$ along the length of the rectangle which puts the ant $3 \mathrm{~cm}$ from the closest vertex.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-262.jpg?height=355&width=1764&top_left_y=240&top_left_x=175)

ANSWER: (A)