Use the areas of the two identical squares to explain why \(5^2+12^2=13^2\) without doing any calculations.

2 decomposed squares. On left square is decomposed into square with side length = 5, 2 rectangles with sides = 12 and 5, and square with side length = 12. On right square is decomposed into a square with side length = 13 and 4 right triangles with sides = 5, 12, 13.

Find the exact value of each variable that represents a side length in a right triangle.

5 right triangles. Top left, Side lengths are: h, 8, 10. Top center, Side lengths are: 6, k, 6 point 5. Top right, side lengths are: 2, m, 5. Bottom left, Side lengths are: square root 10, n, 10. Bottom right, Side lengths are: p, square root 68, square root 85.

In each part, \(a\) and \(b\) represent the length of a leg of a right triangle, and \(c\) represents the length of its hypotenuse. Find the unknown length, given the other two lengths.

1. \(a=3, b=1, c={?}\)
2. \(a={?}, b=2, c=\sqrt{29}\)
3. \(a=\sqrt5, b=\sqrt7, c={?}\)
4. \(a=\sqrt8, b={?}, c=\sqrt{13}\)
5. \(a={?}, b=\sqrt{13}, c=4\)

In 2015, there were roughly \(1 \times 10^6\) high school football players and \(2 \times 10^3\) professional football players in the United States. About how many times more high school football players are there? Explain how you know.

Each number is between which two consecutive integers?

1. \(\sqrt{10}\)
2. \(\sqrt{54}\)
3. \(\sqrt{18}\)
4. \(\sqrt{99}\)
5. \(\sqrt{41}\)