To solve for the length of the longer leg [tex]\( f \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex] for a right triangle where the hypotenuse is three times the length of the shorter leg, we need to follow these steps:
1. Define variables:
- Let [tex]\( a \)[/tex] be the length of the shorter leg.
- Let [tex]\( h \)[/tex] be the hypotenuse.
- Since [tex]\( h \)[/tex] is given to be three times the length of the shorter leg, we have [tex]\( h = 3a \)[/tex].
2. Use the Pythagorean Theorem:
[tex]\[
a^2 + b^2 = h^2
\][/tex]
Here, [tex]\( b \)[/tex] is the longer leg we are solving for.
3. Substitute [tex]\( h = 3a \)[/tex] into the Pythagorean theorem:
[tex]\[
a^2 + b^2 = (3a)^2
\][/tex]
[tex]\[
a^2 + b^2 = 9a^2
\][/tex]
4. Solve for [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = 9a^2 - a^2
\][/tex]
[tex]\[
b^2 = 8a^2
\][/tex]
5. Taking the square root of both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \sqrt{8a^2}
\][/tex]
[tex]\[
b = a\sqrt{8}
\][/tex]
[tex]\[
b = 2a\sqrt{2}
\][/tex]
6. Substitute [tex]\( a = \frac{h}{3} \)[/tex] to express [tex]\( b \)[/tex] in terms of [tex]\( h \)[/tex]:
[tex]\[
b = 2 \left( \frac{h}{3} \right) \sqrt{2}
\][/tex]
[tex]\[
b = \frac{2h\sqrt{2}}{3}
\][/tex]
Thus, the length of the longer leg [tex]\( f \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex] is [tex]\( \frac{2\sqrt{2}}{3}h \)[/tex].
So, the answer is:
[tex]\[ f = \frac{2\sqrt{2}}{3}h \][/tex]
