To find the length of the longer leg [tex]\( l \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex], let's start by understanding the given information.
1. Let the shorter leg be denoted by [tex]\( a \)[/tex].
2. The hypotenuse, [tex]\( h \)[/tex], is three times the shorter leg, [tex]\( a \)[/tex]. Therefore, [tex]\( h = 3a \)[/tex].
3. Let the longer leg be denoted by [tex]\( l \)[/tex].
Using the Pythagorean Theorem for a right triangle:
[tex]\[ h^2 = a^2 + l^2 \][/tex]
Since the hypotenuse [tex]\( h \)[/tex] is [tex]\( 3a \)[/tex], substitute [tex]\( h = 3a \)[/tex] into the Pythagorean Theorem:
[tex]\[ (3a)^2 = a^2 + l^2 \][/tex]
[tex]\[ 9a^2 = a^2 + l^2 \][/tex]
Rearrange this equation to isolate [tex]\( l^2 \)[/tex]:
[tex]\[ 9a^2 - a^2 = l^2 \][/tex]
[tex]\[ 8a^2 = l^2 \][/tex]
To find [tex]\( l \)[/tex], take the square root of both sides:
[tex]\[ l = \sqrt{8a^2} \][/tex]
[tex]\[ l = \sqrt{8} \cdot a \][/tex]
[tex]\[ l = 2\sqrt{2} \cdot a \][/tex]
Now express [tex]\( a \)[/tex] in terms of [tex]\( h \)[/tex]. Since [tex]\( h = 3a \)[/tex]:
[tex]\[ a = \frac{h}{3} \][/tex]
Substitute [tex]\( a \)[/tex] back into the expression for [tex]\( l \)[/tex]:
[tex]\[ l = 2\sqrt{2} \cdot \frac{h}{3} \][/tex]
[tex]\[ l = \frac{2\sqrt{2}}{3} \cdot h \][/tex]
Thus, the length of the longer leg [tex]\( l \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex] is:
[tex]\[ l = \frac{2\sqrt{2}}{3} \cdot h \][/tex]
In simplest form, the equation is:
[tex]\[ l = 2\sqrt{2} \cdot \frac{h}{3} \][/tex]
So, the answer in the required format is:
[tex]\[ I = 2 \sqrt{2} h / 3 \][/tex]
