Two buses P and Q start their journeys from a common bus depot to different destinations. Bus P travels 12 km south to reach point 1, then turns left and travels 13 km to point 2. Next, it turns right and travels 14 km to point 3, then turns left and travels 18 km to point 4. Finally, it turns left and travels 9 km to reach its final stop at point 5.

Bus Q travels 16 km east from the depot to point 6, then turns right and travels 11 km to point 7. It then turns left and travels 22 km to point 8, turns left again and travels 14 km to point 9, and finally turns left and travels 39 km to reach its final stop at point 10.

What is the shortest distance between point 1 and point 6?

Question

Two buses P and Q start their journeys from a common bus depot to different destinations. Bus P travels 12 km south to reach point 1, then turns left and travels 13 km to point 2. Next, it turns right and travels 14 km to point 3, then turns left and travels 18 km to point 4. Finally, it turns left and travels 9 km to reach its final stop at point 5.

Bus Q travels 16 km east from the depot to point 6, then turns right and travels 11 km to point 7. It then turns left and travels 22 km to point 8, turns left again and travels 14 km to point 9, and finally turns left and travels 39 km to reach its final stop at point 10.

What is the shortest distance between point 1 and point 6?

Accepted Answer

**Solution:**

1. First, determine the coordinates of point 1 and point 6 relative to the bus depot, which we consider as origin (0,0).

- Bus P travels 12 km south to point 1.
  So, point 1 coordinates: (0, -12)

- Bus Q travels 16 km east to point 6.
  So, point 6 coordinates: (16, 0)

2. To find the shortest distance between point 1 and point 6, use the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$:

$$
	ext{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

3. Substitute the values:

$$
= \sqrt{(16 - 0)^2 + (0 - (-12))^2} = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20 	ext{ km}
$$

**Answer:** 20 km

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