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A printer creates a right triangular card where the hypotenuse, [tex]\( h \)[/tex], is three times as long as the shorter leg. What is the length of the longer leg, [tex]\( l \)[/tex], in terms of [tex]\( h \)[/tex]? Write the answer in simplest form.

[tex]\[ l = \frac{a \sqrt{b} h}{c} \][/tex]

Sagot :

To find the length of the longer leg [tex]\( I \)[/tex] in terms of the hypotenuse [tex]\( h \)[/tex], given that the hypotenuse is three times the shorter leg, we can use Pythagoras' theorem. Let's go step-by-step:

1. Let the shorter leg be denoted by [tex]\( a \)[/tex]. According to the problem, the hypotenuse [tex]\( h \)[/tex] is three times the shorter leg, so we have:
[tex]\[ h = 3a \][/tex]

2. The relationship between the sides of a right triangle can be described using Pythagoras' theorem:
[tex]\[ h^2 = a^2 + I^2 \][/tex]
Substituting [tex]\( h = 3a \)[/tex] into this equation, we get:
[tex]\[ (3a)^2 = a^2 + I^2 \][/tex]
[tex]\[ 9a^2 = a^2 + I^2 \][/tex]

3. Simplifying the equation:
[tex]\[ 9a^2 - a^2 = I^2 \][/tex]
[tex]\[ 8a^2 = I^2 \][/tex]

4. Taking the square root of both sides to solve for [tex]\( I \)[/tex]:
[tex]\[ I = \sqrt{8a^2} \][/tex]
[tex]\[ I = \sqrt{8} \cdot a \][/tex]
[tex]\[ I = 2\sqrt{2} \cdot a \][/tex]

5. We already know that [tex]\( h = 3a \)[/tex], so we can express [tex]\( a \)[/tex] in terms of [tex]\( h \)[/tex]:
[tex]\[ a = \frac{h}{3} \][/tex]

6. Substituting [tex]\( a = \frac{h}{3} \)[/tex] back into the equation for [tex]\( I \)[/tex]:
[tex]\[ I = 2\sqrt{2} \cdot \frac{h}{3} \][/tex]
[tex]\[ I = \frac{2\sqrt{2}}{3} \cdot h \][/tex]

Thus, the length of the longer leg [tex]\( I \)[/tex] in terms of [tex]\( h \)[/tex] is:
[tex]\[ I = \frac{2 \sqrt{2} h}{3} \][/tex]

Therefore, the correct values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] to plug into the given equation [tex]\( I = \frac{a \sqrt{b} h}{c} \)[/tex] are:
[tex]\[ a = 2, \quad b = 2, \quad c = 3 \][/tex]

So,
[tex]\[ I = \frac{2 \sqrt{2} h}{3} \][/tex]