To solve for the length of the longer leg in terms of [tex]\( m \)[/tex], start by using the known relationship involving the hypotenuse and legs in a right triangle along with the Pythagorean theorem:
1. Identify the known sides:
- The shorter leg is [tex]\( m \)[/tex].
- The hypotenuse [tex]\( h \)[/tex] is [tex]\( 3m \)[/tex].
2. Use the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs of the triangle, and [tex]\( c \)[/tex] is the hypotenuse.
3. Substitute the given values:
[tex]\[ m^2 + b^2 = (3m)^2 \][/tex]
4. Solve for [tex]\( b \)[/tex]:
[tex]\[ m^2 + b^2 = 9m^2 \][/tex]
[tex]\[ b^2 = 9m^2 - m^2 \][/tex]
[tex]\[ b^2 = 8m^2 \][/tex]
[tex]\[ b = \sqrt{8m^2} \][/tex]
[tex]\[ b = \sqrt{8} \cdot \sqrt{m^2} \][/tex]
[tex]\[ b = 2\sqrt{2} \cdot m \][/tex]
Thus, the length of the longer leg [tex]\( b \)[/tex] is [tex]\( 2\sqrt{2} \cdot m \)[/tex].
So, the correct answer is:
[tex]\[ 2\sqrt{2} \cdot m \][/tex]
