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The Pythagorean Theorem is a fundamental principle in geometry that applies to right-angled triangles. In simple terms, it states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is expressed as \( a^2 + b^2 = c^2 \).

In our exercise, the quarterback's movement creates two legs of a right triangle. His backward run is one leg, and his sideways run is the other leg. By applying the Pythagorean Theorem, we can calculate the length of the diagonal path he takes, effectively uncovering the third side of the triangle, or the hypotenuse, representing his resultant displacement from the start position after the runs.

When we talk about vector displacement, we're referring to a vector that gives us both the magnitude and direction of a movement from an initial to a final position. It doesn't matter what path was actually taken; only the start and end points are relevant. In a two-dimensional plane, like a football field, displacements can be represented as vectors, where each movement has both a direction and a distance associated with it.

In our example, the quarterback's backward and sideways runs, as well as his forward pass, are vector displacements. Combined, they give us the total resultant displacement. The importance of vector displacement lies in its ability to summarize complex movements into a single vector that accurately describes the overall change in position.

The right triangle is a powerful tool in understanding movements that involve change in two perpendicular directions. When the legs of the triangle represent movements that are at right angles to each other, such as the quarterback's backward and sideways runs, or the sideways run and the forward pass, the resulting hypotenuse represents the shortest path between the starting and ending points of these movements.

This right triangle structure allows us to use the Pythagorean Theorem to find the precise value of this hypotenuse, which is vital for calculating the true resultant displacement of an object. The use of right triangles is prevalent in various fields, including physics, engineering, and navigation, making it an essential concept for problem-solving in real-world applications.