Understanding Kartick's Journey and Path

To find the shortest distance between point H and point F, we first need to carefully trace Kartick's path from the starting point A, noting down each turn and the distance covered. This type of problem often involves visualizing the movement and then applying geometric principles, specifically the Pythagorean theorem for the shortest distance between two points in a plane.

Let's break down Kartick's movement step-by-step to understand the complete journey:

• Starting Point: Kartick begins his journey from point A.
• First Movement: He travels 30 km to the south and reaches point B.
• Second Movement: From point B, he makes a left turn. Since he was moving South, a left turn means he now moves East. He covers 50 km to reach point C.
• Third Movement: From point C, he makes a right turn. As he was moving East, a right turn means he now moves South. He travels 60 km, and this location is point D.
• Fourth Movement: From point D, he makes a left turn. Being in a Southward direction, a left turn means he now moves East. He covers 50 km to reach point E.
• Fifth Movement: From point E, he makes a right turn. Since he was moving East, a right turn means he now moves South. He travels 20 km, and this location is point F.
• Sixth Movement: From point F, he makes a right turn. Moving South, a right turn means he now moves West. He travels 150 km, reaching point G.
• Seventh Movement: From point G, he makes a right turn. As he was moving West, a right turn means he now moves North. He covers 80 km, and this is point H, where he finally stops.

Calculating Relative Positions of Key Points

To accurately determine the shortest distance between point H and point F, it is beneficial to represent the positions of these points using a coordinate system. Let's assume the starting point A is at the origin \((0,0)\). We will track the X-coordinate (East/West) and Y-coordinate (North/South) for each significant point in Kartick's journey. Positive X will represent East, negative X will represent West. Positive Y will represent North, negative Y will represent South.

From our detailed coordinate tracking, we have the precise locations for the points we need to analyze:

• Point F coordinates: \((x_F, y_F) = (100, -110)\)
• Point H coordinates: \((x_H, y_H) = (-50, -30)\)

Shortest Distance Calculation between H and F

The shortest distance between any two points in a two-dimensional plane can be calculated using the distance formula, which is directly derived from the Pythagorean theorem. If we have two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(D\) between them is given by:

\[ D = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 } \]

Let's apply this formula using the coordinates of point F \((100, -110)\) and point H \((-50, -30)\).

Here, \(x_1 = 100\), \(y_1 = -110\) (for point F) and \(x_2 = -50\), \(y_2 = -30\) (for point H).

Substituting these values into the distance formula: \[ D_{HF} = \sqrt{ (-50 - 100)^2 + (-30 - (-110))^2 } \] \[ D_{HF} = \sqrt{ (-150)^2 + (-30 + 110)^2 } \] \[ D_{HF} = \sqrt{ (-150)^2 + (80)^2 } \]

Now, we calculate the squares of the differences: \[ (-150)^2 = 22500 \] \[ (80)^2 = 6400 \]

Adding these values: \[ D_{HF} = \sqrt{ 22500 + 6400 } \] \[ D_{HF} = \sqrt{ 28900 } \]

Finally, we find the square root of 28900: We know that \(17^2 = 289\), so \(\sqrt{289} = 17\). Thus, \(\sqrt{28900} = \sqrt{289 \times 100} = \sqrt{289} \times \sqrt{100} = 17 \times 10 = 170\).

Therefore, the shortest distance between point H and point F is 170 km.