Mr X runs a marathon starting from the point A he runs 5 km towards north and reach point B, then turns right runs 6 km reach point C, then turn right runs 8 km and reach point D, then turns right runs 10 km and reach point E, then turns right runs 6 km and reach point F, then turns right runs 1 km and reach point G, then turn rights runs 3 km and reach point H.
The shortest distance between point D and G is ?

Options

A. `2sqrt13`
B. `sqrt13`
C. `3sqrt13`
D. 13

Answer

To find the shortest distance between point D and point G, we can follow these steps:

### Step 1: Understand the Path
Mr. X starts at point A and follows a specific path:
- From A to B: 5 km North
- From B to C: 6 km East
- From C to D: 8 km South
- From D to E: 10 km West
- From E to F: 6 km South
- From F to G: 1 km East
- From G to H: 3 km South

### Step 2: Plot the Points
Let's plot the points based on the directions:
- A (0, 0)
- B (0, 5) [5 km North]
- C (6, 5) [6 km East]
- D (6, -3) [8 km South from C (5 - 8 = -3)]
- E (-4, -3) [10 km West from D (6 - 10 = -4)]
- F (-4, -9) [6 km South from E (-3 - 6 = -9)]
- G (-3, -9) [1 km East from F (-4 + 1 = -3)]

### Step 3: Calculate the Coordinates
We have the coordinates:
- D (6, -3)
- G (-3, -9)

### Step 4: Use the Distance Formula
To find the shortest distance (GD) between points D and G, we can use the distance formula:
\[ GD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where:
- \( (x_1, y_1) \) are the coordinates of point D (6, -3)
- \( (x_2, y_2) \) are the coordinates of point G (-3, -9)

### Step 5: Substitute the Coordinates
Substituting the coordinates into the formula:
\[ GD = \sqrt{((-3) - 6)^2 + ((-9) - (-3))^2} \]
\[ = \sqrt{(-9)^2 + (-6)^2} \]
\[ = \sqrt{81 + 36} \]
\[ = \sqrt{117} \]

### Step 6: Simplify the Result
The distance can be simplified:
\[ GD = \sqrt{9 \times 13} = 3\sqrt{13} \]

### Final Answer
Thus, the shortest distance between point D and point G is:
\[ \text{GD} = 3\sqrt{13} \]