To find the length of the missing leg \( a \) of a right triangle, given that \( b = 13 \) inches and \( c = 12 \) inches, we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
In this case, \( b \) is one leg of the triangle and \( c \) is the hypotenuse. Let's plug in the given values:
\[ a^2 + 13^2 = 12^2 \]
Calculating the squares:
\[ a^2 + 169 = 144 \]
Now, we solve for \( a^2 \):
\[ a^2 = 144 - 169 \] \[ a^2 = -25 \]
At this point, we see that \( a^2 \) cannot be negative, which indicates that the lengths given do not correspond to a valid right triangle, given the definition of a right triangle where the hypotenuse must be the longest side.
Thus, we conclude there is a mistake in the provided values as \( c \) (hypotenuse) should be greater than either leg \( a \) or \( b \), but here, \( 12 < 13 \).
If you're looking for \( a \) in the context where \( c \) is the hypotenuse, you might need to confirm the values of \( b \) and \( c \). If your setup is reversed and you meant \( c = 13 \) as the hypotenuse, then the equation would look like:
\[ a^2 + 12^2 = 13^2 \]
Calculating it in that scenario:
\[ a^2 + 144 = 169 \] \[ a^2 = 169 - 144 \] \[ a^2 = 25 \] \[ a = \sqrt{25} = 5 \]
So, if \( c = 13 \) and \( b = 12 \), then the length of the missing leg \( a \) is \( 5 \) inches.
Please clarify the values if this doesn't resolve your question!