To determine the length of the longer leg [tex]\( I \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex], follow these steps:
1. Understand the problem: We know the hypotenuse ([tex]\( h \)[/tex]) is three times the shorter leg. Let’s denote the shorter leg as [tex]\( a \)[/tex].
2. Relation between hypotenuse and shorter leg: Given that [tex]\( h = 3a \)[/tex].
3. Apply the Pythagorean theorem: According to the theorem, for a right triangle:
[tex]\[
h^2 = a^2 + l^2
\][/tex]
where [tex]\( l \)[/tex] is the longer leg.
4. Substitute the hypotenuse:
[tex]\[
(3a)^2 = a^2 + l^2
\][/tex]
which simplifies to:
[tex]\[
9a^2 = a^2 + l^2
\][/tex]
5. Solve for [tex]\( l^2 \)[/tex]:
[tex]\[
l^2 = 9a^2 - a^2 = 8a^2
\][/tex]
6. Express [tex]\( l \)[/tex] in terms of [tex]\( a \)[/tex]:
[tex]\[
l = \sqrt{8a^2} = \sqrt{8} \cdot a = 2\sqrt{2} \cdot a
\][/tex]
7. Substitute [tex]\( a \)[/tex] in terms of [tex]\( h \)[/tex]: Since [tex]\( a = \frac{h}{3} \)[/tex], substitute this into the expression for [tex]\( l \)[/tex]:
[tex]\[
l = 2\sqrt{2} \cdot \frac{h}{3}
\][/tex]
8. Simplify the expression:
[tex]\[
l = \frac{2\sqrt{2} \cdot h}{3}
\][/tex]
Thus, the length of the longer leg [tex]\( I \)[/tex] can be written as:
[tex]\[
I = \frac{2\sqrt{2}h}{3}
\][/tex]
In the format [tex]\( I = \frac{a \sqrt{b} h}{c} \)[/tex], the values are:
[tex]\[
a = 2, \quad b = 2, \quad c = 3
\][/tex]
So the final answer is:
[tex]\[
I = \frac{2 \sqrt{2} h}{3}
\][/tex]
