To determine the length of the longer leg [tex]\( I \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex] for a right triangular card where the hypotenuse [tex]\( h \)[/tex] is three times as long as the shorter leg, we follow these steps:
1. Let the shorter leg be [tex]\( x \)[/tex].
2. Given that the hypotenuse [tex]\( h \)[/tex] is three times the length of the shorter leg, we have:
[tex]\[
h = 3x
\][/tex]
3. According to the Pythagorean theorem for a right triangle:
[tex]\[
(shorter \, leg)^2 + (longer \, leg)^2 = (hypotenuse)^2
\][/tex]
Therefore:
[tex]\[
x^2 + (I)^2 = (3x)^2
\][/tex]
4. Simplify the equation:
[tex]\[
x^2 + I^2 = 9x^2
\][/tex]
5. Rearrange to solve for the longer leg [tex]\( I \)[/tex]:
[tex]\[
I^2 = 9x^2 - x^2
\][/tex]
[tex]\[
I^2 = 8x^2
\][/tex]
[tex]\[
I = \sqrt{8x^2}
\][/tex]
6. Simplify the expression under the square root:
[tex]\[
I = x \sqrt{8}
\][/tex]
[tex]\[
I = x \cdot 2\sqrt{2}
\][/tex]
7. Substitute [tex]\( x = \frac{h}{3} \)[/tex] back into the expression:
[tex]\[
I = \frac{h}{3} \cdot 2\sqrt{2}
\][/tex]
[tex]\[
I = \frac{2\sqrt{2} h}{3}
\][/tex]
Therefore, the length of the longer leg [tex]\( I \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex] is:
[tex]\[
I = \frac{2\sqrt{2} h}{3}
\][/tex]
In the expression [tex]\( \frac{a \sqrt{b} h}{c} \)[/tex], we have:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = 3 \)[/tex]
So, the correct answer is:
[tex]\[
I = \frac{2 \sqrt{2} h}{3}
\][/tex]
