1. A ladder is leaning against a wall. The base of the ladder is 6 feet away from the wall, and the ladder is 8 feet long. How far up the wall does the ladder reach?

Solution: Using the Pythagorean theorem, we can find the height of the ladder on the wall. Let’s call the height “h”. According to the theorem, the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides. So, we have:

h^2 + 6^2 = 8^2 h^2 + 36 = 64 h^2 = 64 - 36 h^2 = 28 h = √28 h ≈ 5.29

Therefore, the ladder reaches approximately 5.29 feet up the wall.

1. A right-angled triangle has one side measuring 5 cm and another side measuring 12 cm. What is the length of the hypotenuse?

Solution: Let’s call the length of the hypotenuse “c”. Using the Pythagorean theorem, we have:

5^2 + 12^2 = c^2 25 + 144 = c^2 169 = c^2 c = √169 c = 13

Therefore, the length of the hypotenuse is 13 cm.

1. A baseball diamond is a square with sides measuring 90 feet. How far is it from home plate to second base?

Solution: The distance from home plate to second base forms the hypotenuse of a right-angled triangle. Let’s call this distance “c”. Using the Pythagorean theorem, we have:

c^2 = 90^2 + 90^2 c^2 = 8100 + 8100 c^2 = 16200 c = √16200 c ≈ 127.28

Therefore, it is approximately 127.28 feet from home plate to second base.