jackson runs a straight route 40 yards up the sideline before turning around and catching a pass thrown by michael vick. on the opposing team. a defender who started 20 yards across the field from jackson saw the play setup and ran a slant toward jackson what was the distance the defender had to run to get to the spot where jackson caught the ball?​

To determine the distance the defender had to run to get to the spot where Jackson caught the ball, we can use the Pythagorean theorem. This theorem can be applied because the paths that Jackson and the defender run form a right triangle.

The right triangle is formed by: - The 40 yards that Jackson runs straight up the sideline (the first leg of the triangle), - The 20 yards across the field the defender starts from (the second leg of the triangle), - The diagonal path the defender runs toward the spot Jackson caught the ball (the hypotenuse of the triangle).

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is expressed as c² = a² + b².

In this scenario: - a = 40 yards (Jackson's run) - b = 20 yards (the defender's starting distance across the field)

Now we calculate the defender's distance: c² = a² + b² c² = 40² + 20² c² = 1600 + 400 c² = 2000

To find c (the distance the defender ran), we take the square root of 2000:

c = √2000 c ≈ 44.7 yards

Therefore, the defender had to run approximately 44.7 yards to get to the spot where Jackson caught the ball.

Extra: The Pythagorean theorem is an important principle in geometry, especially when dealing with right-angled triangles. It is widely used in various fields such as construction, navigation, and even sports, as we see in this example.

Understanding the Pythagorean theorem is not only about knowing how to calculate the lengths of sides but also about visualizing and identifying when a scenario forms a right triangle. Whenever we have a situation with perpendicular distances, like in this football scenario, we can often apply this theorem.

Moreover, the theorem demonstrates the relationship between the sides of a right triangle, illustrating that the hypotenuse is always the longest side. In more advanced mathematics, we see that the Pythagorean theorem is a special case of the more general Law of Cosines, which applies to any triangle, not just right-angled ones.

For a school student, learning about the Pythagorean theorem starts with the recognition of right triangles and the understanding that it can be used to find a missing length when the lengths of the other two sides are known. Practical examples like this one in sports can help students better understand and remember the theorem's application.