To determine which sets of side measurements can form a right triangle, we can use the Pythagorean theorem, which states that for a right triangle with side lengths \( a \), \( b \), and hypotenuse \( c \) (where \( c \) is the longest side):
\[ a^2 + b^2 = c^2 \]
We will examine each set of side measurements:
-
Set: 14, 5, 15
- Here, \( c = 15 \), \( a = 14 \), \( b = 5 \)
- Calculate: \( a^2 + b^2 = 14^2 + 5^2 = 196 + 25 = 221 \)
- Calculate: \( c^2 = 15^2 = 225 \)
- Since \( 221 \neq 225 \), this set does not form a right triangle.
-
Set: 3, 4, 5
- Here, \( c = 5 \), \( a = 4 \), \( b = 3 \)
- Calculate: \( a^2 + b^2 = 4^2 + 3^2 = 16 + 9 = 25 \)
- Calculate: \( c^2 = 5^2 = 25 \)
- Since \( 25 = 25 \), this set does form a right triangle.
-
Set: 9, 14, 16
- Here, \( c = 16 \), \( a = 14 \), \( b = 9 \)
- Calculate: \( a^2 + b^2 = 14^2 + 9^2 = 196 + 81 = 277 \)
- Calculate: \( c^2 = 16^2 = 256 \)
- Since \( 277 \neq 256 \), this set does not form a right triangle.
-
Set: 5, 2, 7
- Here, \( c = 7 \), \( a = 5 \), \( b = 2 \)
- Calculate: \( a^2 + b^2 = 5^2 + 2^2 = 25 + 4 = 29 \)
- Calculate: \( c^2 = 7^2 = 49 \)
- Since \( 29 \neq 49 \), this set does not form a right triangle.
In conclusion, the only set of side measurements that can form a right triangle is:
3, 4, 5.