The Pythagorean theorem in three dimensions practice answers

Answer 1

Here are some practice answers for applying the Pythagorean theorem in three dimensions:

1. Find the length of the hypotenuse in a right-angled triangle with sides measuring 3 units and 4 units.

Using the Pythagorean theorem, we can calculate:
c^2 = 3^2 + 4^2
c^2 = 9 + 16
c^2 = 25
c = √25
c = 5

Therefore, the length of the hypotenuse is 5 units.

2. A rectangular prism has dimensions 5 units, 12 units, and 13 units. Find the length of the longest diagonal.

We can use the Pythagorean theorem twice to find the length of the longest diagonal:
Diagonal 1 = √(5^2 + 12^2) = √(25 + 144) = √169 = 13
Diagonal 2 = √(5^2 + 13^2) = √(25 + 169) = √194 ≈ 13.928

Therefore, the length of the longest diagonal is approximately 13.928 units.

3. In a right-angled triangle, one side measures 8 units and the hypotenuse measures 10 units. Find the length of the other side.

Using the Pythagorean theorem, we can calculate:
10^2 = 8^2 + x^2
100 = 64 + x^2
x^2 = 100 - 64
x^2 = 36
x = √36
x = 6

Therefore, the length of the other side is 6 units.