To solve for the length of the longer leg [tex]\( l \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex] of a right triangle where the hypotenuse is three times as long as the shorter leg, follow these steps:
1. Let the shorter leg be [tex]\( x \)[/tex]. According to the problem, the hypotenuse [tex]\( h \)[/tex] is three times the shorter leg. Therefore, we have:
[tex]\[
h = 3x
\][/tex]
2. Use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This can be written as:
[tex]\[
h^2 = x^2 + l^2
\][/tex]
3. Substitute [tex]\( h = 3x \)[/tex] into the Pythagorean theorem equation:
[tex]\[
(3x)^2 = x^2 + l^2
\][/tex]
Simplify:
[tex]\[
9x^2 = x^2 + l^2
\][/tex]
4. Solve for [tex]\( l^2 \)[/tex]:
[tex]\[
9x^2 - x^2 = l^2
\][/tex]
[tex]\[
8x^2 = l^2
\][/tex]
5. Take the positive square root of both sides to solve for [tex]\( l \)[/tex]:
[tex]\[
l = \sqrt{8x^2}
\][/tex]
[tex]\[
l = x\sqrt{8}
\][/tex]
6. Substitute [tex]\( x = \frac{h}{3} \)[/tex] (from [tex]\( h = 3x \)[/tex]) back into the equation for [tex]\( l \)[/tex]:
[tex]\[
l = \left(\frac{h}{3}\right) \sqrt{8}
\][/tex]
7. Simplify the expression:
[tex]\[
l = \frac{h \sqrt{8}}{3}
\][/tex]
So the length of the longer leg [tex]\( l \)[/tex] in terms of [tex]\( h \)[/tex] is:
[tex]\[
l = \frac{h \sqrt{8}}{3}
\][/tex]
Here’s how the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] fit into your given format [tex]\( l = \frac{a \sqrt{b} h}{c} \)[/tex]:
- [tex]\( a = 1 \)[/tex] (since [tex]\( a \)[/tex] is multiplying [tex]\( \sqrt{8} \)[/tex])
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = 3 \)[/tex]
Thus, the correct expression is:
[tex]\[
l = \frac{1 \sqrt{8} h}{3}
\][/tex]
