To determine the length of the longer leg [tex]\( f \)[/tex] of a right triangle where the hypotenuse [tex]\( h \)[/tex] is three times the length of the shorter leg [tex]\( x \)[/tex], we can follow these steps:
1. Establish the relationship between the hypotenuse and the shorter leg:
Since the hypotenuse [tex]\( h \)[/tex] is three times the shorter leg, we have:
[tex]\[
h = 3x
\][/tex]
This implies that the shorter leg [tex]\( x \)[/tex] can be expressed as:
[tex]\[
x = \frac{h}{3}
\][/tex]
2. Use the Pythagorean theorem:
According to the Pythagorean theorem for a right triangle, the square of the hypotenuse [tex]\( h \)[/tex] is equal to the sum of the squares of the legs:
[tex]\[
h^2 = x^2 + f^2
\][/tex]
3. Substitute [tex]\( x \)[/tex] in terms of [tex]\( h \)[/tex] into the equation:
Substituting [tex]\( x = \frac{h}{3} \)[/tex] into the Pythagorean theorem equation, we get:
[tex]\[
h^2 = \left( \frac{h}{3} \right)^2 + f^2
\][/tex]
4. Simplify the equation:
[tex]\[
h^2 = \frac{h^2}{9} + f^2
\][/tex]
5. Solve for [tex]\( f^2 \)[/tex]:
To isolate [tex]\( f^2 \)[/tex], subtract [tex]\(\frac{h^2}{9}\)[/tex] from both sides of the equation:
[tex]\[
h^2 - \frac{h^2}{9} = f^2
\][/tex]
6. Simplify the left-hand side:
[tex]\[
h^2 - \frac{h^2}{9} = \frac{9h^2}{9} - \frac{h^2}{9} = \frac{8h^2}{9}
\][/tex]
7. Therefore, [tex]\( f^2 \)[/tex]:
[tex]\[
f^2 = \frac{8h^2}{9}
\][/tex]
8. Take the square root of both sides to solve for [tex]\( f \)[/tex]:
[tex]\[
f = \sqrt{\frac{8h^2}{9}} = \frac{\sqrt{8h^2}}{3} = \frac{2\sqrt{2}h}{3}
\][/tex]
So, the length of the longer leg [tex]\( f \)[/tex] in terms of [tex]\( h \)[/tex] is:
[tex]\[
f = \frac{2\sqrt{2}h}{3}
\][/tex]
So, the correct values for [tex]\( a, b \)[/tex], and [tex]\( c \)[/tex] are:
[tex]\[
\boxed{2\sqrt{2}, 2, 3}
\][/tex]
