To determine whether a right triangle can be formed with sides \( a = 6 \), \( b = 10 \), and \( c = 12 \), we will use the Converse of the Pythagorean Theorem, which states:
If \( c \) is the longest side, then a triangle is a right triangle if \( a^2 + b^2 = c^2 \).
In this case, we have:
- \( a = 6 \)
- \( b = 10 \)
- \( c = 12 \)
First, we calculate \( a^2 \) and \( b^2 \):
\[ a^2 = 6^2 = 36 \] \[ b^2 = 10^2 = 100 \]
Now, we sum \( a^2 \) and \( b^2 \):
\[ a^2 + b^2 = 36 + 100 = 136 \]
Next, we calculate \( c^2 \):
\[ c^2 = 12^2 = 144 \]
Now, we compare \( a^2 + b^2 \) with \( c^2 \):
\[ a^2 + b^2 = 136 \quad \text{and} \quad c^2 = 144 \]
Since \( 136 \neq 144 \), we find that \( a^2 + b^2 \neq c^2 \).
Thus, the correct response is:
No, a right triangle cannot be formed because \( 6^2 + 10^2 \neq 12^2 \).