To find the length of the longer leg, [tex]\( l \)[/tex], in terms of the hypotenuse, [tex]\( h \)[/tex], given that the hypotenuse is three times as long as the shorter leg, follow these steps:
1. Identify the sides of the triangle:
- Let the shorter leg be [tex]\( s \)[/tex].
- The hypotenuse is [tex]\( h \)[/tex], and it is given that [tex]\( h = 3s \)[/tex]. This gives us [tex]\( s = \frac{h}{3} \)[/tex].
- Let the longer leg be [tex]\( l \)[/tex].
2. Apply the Pythagorean theorem:
[tex]\[
s^2 + l^2 = h^2
\][/tex]
Substitute [tex]\( s = \frac{h}{3} \)[/tex] into the equation:
[tex]\[
\left( \frac{h}{3} \right)^2 + l^2 = h^2
\][/tex]
3. Simplify the equation:
[tex]\[
\left( \frac{h}{3} \right)^2 = \frac{h^2}{9}
\][/tex]
Therefore:
[tex]\[
\frac{h^2}{9} + l^2 = h^2
\][/tex]
4. Isolate [tex]\( l^2 \)[/tex]:
[tex]\[
l^2 = h^2 - \frac{h^2}{9}
\][/tex]
Rewrite [tex]\( h^2 \)[/tex] as a common fraction with the same denominator:
[tex]\[
h^2 = \frac{9h^2}{9}
\][/tex]
Thus:
[tex]\[
l^2 = \frac{9h^2}{9} - \frac{h^2}{9} = \frac{8h^2}{9}
\][/tex]
5. Solve for [tex]\( l \)[/tex]:
[tex]\[
l = \sqrt{\frac{8h^2}{9}} = \frac{\sqrt{8h^2}}{3} = \frac{\sqrt{8} \cdot \sqrt{h^2}}{3} = \frac{2\sqrt{2} h}{3}
\][/tex]
The length of the longer leg [tex]\( l \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex] is:
[tex]\[
l = \frac{2\sqrt{2}\sqrt{h}}{3}
\][/tex]
Therefore, the correct values to replace [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] in the expression [tex]\( I = \frac{a \sqrt{b h}}{c} \)[/tex] are:
[tex]\[
I = \frac{2\sqrt{2h}}{3}
\][/tex]
So, [tex]\( a = 2 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 3 \)[/tex].
