Let's break down this spatial reasoning problem step by step to determine the required distances and their difference.

Understanding the Initial Positions

We are given the relative positions of VD Raman's house, a temple, and a church. To make this easier, we can set up a coordinate system. Let's place the Church at the origin (0,0).

• The Church is at (0, 0).
• The temple is 40 m to the north of the church. So, the Temple is at (0, 40).
• VD Raman’s house is 30 m to the left of the temple. Left typically means West, which corresponds to decreasing the x-coordinate. So, the House is at (0 - 30, 40) = (-30, 40).

Mapping VD Raman's Movement to the Gym

VD Raman starts from his house at (-30, 40).

• He walks 50 m along a road which is parallel to the temple-church road. The temple-church road lies along the y-axis (x=0). A road parallel to this passing by his house (-30, 40) would be the line x = -30. Walking 50 m parallel to the temple-church road means moving along this line x = -30, changing the y-coordinate by 50. Starting at y=40, he could move North (y increases) to 40+50=90 or South (y decreases) to 40-50=-10.
• He then turns right and walks 20 m to reach the gym.

Let's consider the direction of the 50 m walk:

• If he walks 50 m North along x=-30, he reaches (-30, 40+50) = (-30, 90). From here, moving North, turning right is turning East (increasing x). Walking 20 m East takes him to (-30+20, 90) = (-10, 90). This would be the Gym's location.
• If he walks 50 m South along x=-30, he reaches (-30, 40-50) = (-30, -10). From here, moving South, turning right is turning West (decreasing x). Walking 20 m West takes him to (-30-20, -10) = (-50, -10). This would be the Gym's location.

We need to determine which case leads to one of the given options. Let's calculate the required distances for the second case, as it usually fits these types of problems unless specified otherwise.

Assuming Case 2: The Gym is at (-50, -10).

Calculating Straight-Line Distances

We need to find the straight distance between VD Raman's house and the church, and the straight distance between the temple and the gym.

Distance 1: House to Church

• House coordinates: (-30, 40)
• Church coordinates: (0, 0)

Using the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\):

Distance (House to Church) \(= \sqrt{(0 - (-30))^2 + (0 - 40)^2}\)

Distance (House to Church) \(= \sqrt{(30)^2 + (-40)^2}\)

Distance (House to Church) \(= \sqrt{900 + 1600}\)

Distance (House to Church) \(= \sqrt{2500}\)

Distance (House to Church) \(= 50\) m.

Distance 2: Temple to Gym

• Temple coordinates: (0, 40)
• Gym coordinates (Case 2): (-50, -10)

Using the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\):

Distance (Temple to Gym) \(= \sqrt{(-50 - 0)^2 + (-10 - 40)^2}\)

Distance (Temple to Gym) \(= \sqrt{(-50)^2 + (-50)^2}\)

Distance (Temple to Gym) \(= \sqrt{2500 + 2500}\)

Distance (Temple to Gym) \(= \sqrt{5000}\) m.

Finding the Difference

The question asks for the difference between the straight distance between VD Raman’s house and the church and the temple and the gym.

Difference \(= |\) Distance (House to Church) - Distance (Temple to Gym) \(|\)

Difference \(= |50 - \sqrt{5000}|\)

Let's calculate the value of \(\sqrt{5000}\):

\(\sqrt{5000} = \sqrt{2500 \times 2} = 50\sqrt{2}\)

Using the approximate value \(\sqrt{2} \approx 1.414\):

\(50\sqrt{2} \approx 50 \times 1.414 = 70.7\) m.

Difference \(\approx |50 - 70.7|\)

Difference \(\approx |-20.7|\)

Difference \(\approx 20.7\) m.

This value matches one of the given options. Therefore, our interpretation of the movement in Case 2 was correct.

The difference between the two distances is approximately 20.7 m.

Revision Table: Key Locations and Distances

Additional Information: Coordinate Geometry and Distance Formula

This problem uses basic concepts from coordinate geometry to solve a spatial reasoning question. The distance formula is a fundamental tool used here.

• Coordinate System: Placing points on a 2D plane using ordered pairs (x, y) allows us to represent locations and movements mathematically.
• Relative Directions: "North" typically means increasing the y-coordinate, "South" means decreasing y, "East" means increasing x, and "West" means decreasing x. "Left" and "Right" depend on the direction of travel. If moving South, right is West. If moving North, right is East.
• Distance Formula: The straight-line distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a 2D plane is given by the formula: \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). This formula is derived directly from the Pythagorean theorem.

Understanding how to translate directional information into coordinate movements and applying the distance formula is crucial for solving such problems.