Understanding the Direction and Distance Problem

This question asks us to calculate the straight-line distance between the child's starting point and the final point where he meets his father. We need to track the child's movements step-by-step and determine his final position relative to the start.

Step-by-Step Analysis of the Child's Movement

Let's break down the child's journey:

1. Initially, the child starts at a point. Let's call this the starting point.
2. He goes 90 feet in the West direction. This is a displacement of 90 feet West from the start.
3. He then turns right. Since he was facing West, turning right means he now faces North. He goes 20 feet in this direction. This is a displacement of 20 feet North from his current position.
4. After this, he turns right again. From facing North, turning right means he now faces East. He walks 30 feet East. This takes him to his uncle's house. This is a displacement of 30 feet East from the previous position.
5. From his uncle's house, he goes 100 feet to his South. This is a displacement of 100 feet South from the uncle's house. Here, he meets his father.

Calculating the Final Position Relative to the Starting Point

To find the final position relative to the start, let's consider the net displacement in the East-West and North-South directions.

• East-West Displacement:
  • Initial movement: 90 feet West (-90 feet East).
  • Third movement: 30 feet East (+30 feet East).
  • Net East-West displacement = $-90 + 30 = -60$ feet East, which is 60 feet West.
• North-South Displacement:
  • Second movement: 20 feet North (+20 feet North).
  • Fifth movement: 100 feet South (-100 feet North).
  • Net North-South displacement = $+20 - 100 = -80$ feet North, which is 80 feet South.

So, the child meets his father at a point that is 60 feet West and 80 feet South from the starting point.

Finding the Straight-Line Distance Using the Pythagorean Theorem

The final position is 60 feet West and 80 feet South of the start. This forms a right-angled triangle where the two shorter sides are 60 feet and 80 feet, and the hypotenuse is the straight-line distance from the start to the final point. We can use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find this distance.

Let the distance be $D$.

According to the Pythagorean theorem:

$$D^2 = (\text{Net West Displacement})^2 + (\text{Net South Displacement})^2$$

$$D^2 = (60 \text{ feet})^2 + (80 \text{ feet})^2$$

$$D^2 = 3600 + 6400$$

$$D^2 = 10000$$

To find $D$, we take the square root of 10000:

$$D = \sqrt{10000}$$

$$D = 100 \text{ feet}$$

Therefore, the child met his father 100 feet from the starting point.

Summary of Movements and Final Displacement

The final position is 60 feet West (-60 East) and 80 feet South (-80 North) from the starting point.

Distance from start = $\sqrt{(-60)^2 + (-80)^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100$ feet.

Revision Table: Key Concepts for Direction and Distance Problems

Additional Information: Solving Direction and Distance MCQs

Solving direction and distance problems often involves visualizing the path or using coordinates. Here are some tips:

• Draw a diagram: Sketching the path helps a lot in understanding the movements and the final position relative to the start.
• Use a coordinate system: Assign the starting point as (0,0). Movements East increase the x-coordinate, West decrease it. Movements North increase the y-coordinate, South decrease it.
• Calculate net displacement: Sum up all movements in the East-West direction and all movements in the North-South direction separately.
• Apply Pythagorean theorem: If you end up with a net displacement in both the horizontal (East-West) and vertical (North-South) directions, the distance from the start is the hypotenuse of the right triangle formed by these net displacements.
• Be careful with turns: "Turning right" or "turning left" changes the direction of movement.

These types of questions test your ability to follow instructions involving directions and apply basic geometry (like the Pythagorean theorem) to find distances.