The doorway of the family room measures 6 feet by 3 feet. What is the length of the diagonal of the doorway?

Let y feet be the length of the diagonal of the doorway.

The height, the width and the diagonal of the doorway form a right-angled triangle, and the diagonal is the hypotenuse.

By Pythagorean theorem :
y² = 6² + 3²
y² = 45
y = √45
y = 3√5
y ≈ 6.7 (to 1 decimal place)

The length of the diagonal of the doorway = 3√5 feet ≈ 6.7 feet

d² = 6² + 3² = 45
d = 3 √5 ft

The diagonal is the hypotinuse of a right triangle. Use the Pythagorean theorem to solve. 6^2. + 3^2. = C^2
36 + 9 = C^2
45 = C^2
C = square root of 45

Use Pythagoras.