Question

a mason is using a right triangle to check a concrete basement floor to ensure that there are 90° angles in each corner. if the sides $overline{ed}$ and $overline{fd}$ measure 3 inches and 4 inches, then the distance between the two endpoints of $overline{ef}$ must be \\(\boxed{}\\) inches to ensure that $\triangle def$ is a right triangle.

Explanation:

Step1: Identify the triangle type

The triangle \( \triangle DEF \) is a right triangle with \( \angle D = 90^\circ \), so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), \( a^2 + b^2 = c^2 \).
Here, \( ED = 3 \) (one leg, \( a = 3 \)) and \( FD = 4 \) (the other leg, \( b = 4 \)), and \( EF \) is the hypotenuse (\( c \)) we need to find.

Step2: Apply the Pythagorean theorem

Substitute \( a = 3 \) and \( b = 4 \) into the formula \( c^2 = a^2 + b^2 \).
\[
c^2 = 3^2 + 4^2
\]
Calculate \( 3^2 = 9 \) and \( 4^2 = 16 \).
\[
c^2 = 9 + 16
\]
\[
c^2 = 25
\]
Take the square root of both sides to find \( c \):
\[
c = \sqrt{25} = 5
\]

Answer:

5