To determine if triangle DEF is a right triangle, we can use the Pythagorean theorem. The sides of the triangle are 50, 40, and 32.
The Pythagorean theorem states that for a triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the longest side), the triangle is a right triangle if:
\[ a^2 + b^2 = c^2 \]
Here, let's assign:
- \( c = 50 \) (the longest side)
- \( a = 40 \)
- \( b = 32 \)
Now we will calculate \( a^2 + b^2 \) and \( c^2 \):
\[ a^2 + b^2 = 40^2 + 32^2 = 1600 + 1024 = 2624 \] \[ c^2 = 50^2 = 2500 \]
Now we compare \( a^2 + b^2 \) and \( c^2 \):
\[ 2624 \neq 2500 \]
Since \( a^2 + b^2 \) does not equal \( c^2 \), triangle DEF is not a right triangle. Therefore, the answer is:
\[ 0 \]