To find the length of the longer leg of a right triangle where the hypotenuse [tex]\( h \)[/tex] is three times as long as the shorter leg, labeled as [tex]\( m \)[/tex], we follow these steps:
1. Identify the sides of the triangle:
- The shorter leg is [tex]\( a = m \)[/tex].
- The hypotenuse is [tex]\( h = 3m \)[/tex].
2. Use the Pythagorean theorem, which states [tex]\( a^2 + b^2 = c^2 \)[/tex], to find the length of the longer leg [tex]\( b \)[/tex]:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Given:
[tex]\[
a = m, \quad c = 3m
\][/tex]
Substitute these into the Pythagorean theorem:
[tex]\[
m^2 + b^2 = (3m)^2
\][/tex]
Simplify the right side:
[tex]\[
m^2 + b^2 = 9m^2
\][/tex]
Isolate [tex]\( b^2 \)[/tex] by subtracting [tex]\( m^2 \)[/tex] from both sides:
[tex]\[
b^2 = 9m^2 - m^2
\][/tex]
Simplify the equation:
[tex]\[
b^2 = 8m^2
\][/tex]
Take the square root of both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \sqrt{8m^2}
\][/tex]
Simplify the square root:
[tex]\[
b = \sqrt{4 \cdot 2 \cdot m^2} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{m^2} = 2 \cdot \sqrt{2} \cdot m = 2\sqrt{2}m
\][/tex]
Therefore, the length of the longer leg in terms of [tex]\( m \)[/tex] is:
[tex]\[
\boxed{2\sqrt{2}m}
\][/tex]
