To solve for the length of leg LS in the right triangle with vertices L, S, and Y, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \):
\[ a^2 + b^2 = c^2 \]
In this case, we have:
- \( YS = 6.3 \) km (one leg of the triangle)
- \( YL = 15.4 \) km (the hypotenuse)
- \( LS \) is the other leg we need to find.
Assigning the lengths to the variables:
- Let \( a = YS = 6.3 \) km
- Let \( c = YL = 15.4 \) km
- Let \( b = LS \) (the unknown leg we want to find)
We will apply the Pythagorean theorem:
\[ YS^2 + LS^2 = YL^2 \]
Plugging in the known values:
\[ (6.3)^2 + LS^2 = (15.4)^2 \]
Calculating \( (6.3)^2 \) and \( (15.4)^2 \):
\[ 6.3^2 = 39.69 \] \[ 15.4^2 = 237.16 \]
Substituting these values into the equation:
\[ 39.69 + LS^2 = 237.16 \]
Now, subtract \( 39.69 \) from both sides to solve for \( LS^2 \):
\[ LS^2 = 237.16 - 39.69 \] \[ LS^2 = 197.47 \]
Next, we take the square root of both sides to find \( LS \):
\[ LS = \sqrt{197.47} \] \[ LS \approx 14.04 \]
Rounding to the nearest tenth gives:
\[ LS \approx 14.0 \text{ km} \]
Thus, the length of \( LS \) is approximately 14.0 km.
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