Understanding Bus Movement and Displacement

Let's break down the bus's journey step by step to figure out the final position relative to shop G.

We can represent the movements using directions and distances. Let's assume the starting point is the origin (0,0) on a coordinate plane, where North is positive Y, South is negative Y, East is positive X, and West is negative X.

• Starts at (0,0).
• Travels 100 m towards the South: Position is (0, -100).
• Takes a right turn (towards East) and travels 300 m: Position is (0 + 300, -100) = (300, -100).
• Again takes a right turn (towards South) and travels 100 m: Position is (300, -100 - 100) = (300, -200). This is shop G.

So, the location of shop G is effectively 300 m East and 200 m South from the starting point.

Now, let's trace the journey from shop G to hospital M:

• Starts from shop G (300, -200).
• Travels 500 m in the North direction: Position is (300, -200 + 500) = (300, 300).
• Finally takes a left turn (towards West) and travels 1200 m: Position is (300 - 1200, 300) = (-900, 300). This is hospital M.

So, the location of hospital M is -900 m in the X direction (West) and 300 m in the Y direction (North) relative to the starting point.

Calculating Position of Hospital M from Shop G

We need to find the position of hospital M relative to shop G. Let's find the displacement in the East-West and North-South directions from G to M.

• Shop G coordinates: (300, -200) relative to the start.
• Hospital M coordinates: (-900, 300) relative to the start.

Displacement in X direction (East-West) = X coordinate of M - X coordinate of G = -900 - 300 = -1200 m.

Since West is the negative X direction, this means the displacement is 1200 m West.

Displacement in Y direction (North-South) = Y coordinate of M - Y coordinate of G = 300 - (-200) = 300 + 200 = 500 m.

Since North is the positive Y direction, this means the displacement is 500 m North.

So, hospital M is 1200 m West and 500 m North of shop G.

Finding the Straight-Line Distance from Shop G to Hospital M

To find the straight-line distance between shop G and hospital M, we can use the Pythagorean theorem, as the movements North and West form the two sides of a right-angled triangle, and the straight-line distance is the hypotenuse.

Let the distance be D.

Using the Pythagorean theorem: $D^2 = (\text{North displacement})^2 + (\text{West displacement})^2$

In this case, North displacement = 500 m and West displacement = 1200 m.

$\text{D}^2 = (500 \text{ m})^2 + (1200 \text{ m})^2$

$\text{D}^2 = 250000 \text{ m}^2 + 1440000 \text{ m}^2$

$\text{D}^2 = 1690000 \text{ m}^2$

$D = \sqrt{1690000 \text{ m}^2}$

$D = 1300 \text{ m}$

The straight-line distance from shop G to hospital M is 1300 m.

Determining the Direction of Hospital M from Shop G

We found that hospital M is 500 m North and 1200 m West of shop G.

When a point is North and West of another point, its direction is North-West.

Summary of Findings

Based on the movements, hospital M is:

• 500 m North of shop G.
• 1200 m West of shop G.

The straight-line distance is 1300 m.

The direction is North-West.