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The Pythagorean theorem is a fundamental concept in geometry that relates to right-angled triangles. It is an essential tool for calculating displacement, especially when dealing with the net movement across two perpendicular directions.
In simple terms, the Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be represented by the equation:

• For a triangle with sides \(a\), \(b\), and hypotenuse \(c\), the theorem is \(a^2 + b^2 = c^2\)

When calculating displacement, like in the truck's journey, this theorem helps us find the shortest path between the starting and ending points. Once you've determined the legs of the triangle formed by the movements, the Pythagorean theorem allows you to calculate the direct line of travel, or the resultant displacement.

Vector addition is a method used to find the resultant vector when you have multiple movement vectors acting in different directions. Think of vectors as arrows; they possess both magnitude and direction.
To add two vectors, you align them tip to tail. The resultant vector, or the net vector, is then drawn from the tail of the first vector to the tip of the last vector. In terms of coordinates:

• Each direction (north-south, east-west) is considered separately.

In the scenario with the delivery truck, vector addition is used to combine the north and south movements, as well as include the eastward movement into a single net displacement vector. The Pythagorean theorem is then used to find the magnitude of this net vector. This practice is crucial for understanding how to combine various directional displacements into one single direct path.

Net movement refers to the overall change in position from the starting point, taking into account all movements in various directions. It involves considering both the magnitude and direction of each movement.
In the truck's example, the net north-south movement was calculated by subtracting the southward displacement from the northward displacement:

• 28 blocks north minus 26 blocks south equals 2 blocks north.

This means the truck's final position in the north-south axis is two blocks north of its starting point. Similarly, since there was 16 blocks movement eastward with no western movement, the net eastward movement is simply 16 blocks east.
By understanding net movement, you can better grasp how to simplify complex paths into clear and easily computable terms, making it easier to apply mathematical techniques like the Pythagorean theorem to find the shortest distance.

A right-angled triangle is a type of triangle where one of its angles is exactly 90 degrees. This geometric shape plays a crucial role in displacement calculations when directions are perpendicular.
When a journey comprises movements in two perpendicular directions - such as north and east - these movements form the legs of a right-angled triangle.

• In the truck's journey, the north-south movement forms one leg (2 blocks), and the eastward movement forms the second leg (16 blocks).

The concept of a right-angled triangle helps break down complex movements into manageable segments that make it possible to apply the Pythagorean theorem. By doing so, one can accurately determine the resultant displacement – the hypotenuse in this scenario. A right-angled triangle provides a simple yet powerful visual framework to solve problems involving multi-directional displacements.