To find the length of \( LS \) in the right triangle \( YSL \), where \( Y \) is the bottom left vertex, \( S \) is the bottom right vertex, and \( L \) is the top vertex, we can use the Pythagorean theorem.
Pythagorean Theorem: \[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides of the right triangle.
In this triangle:
- \( YL \) is one leg (the height) and is given as \( 15.4 , \text{km} \).
- \( YS \) is the other leg (the base) and is given as \( 6.3 , \text{km} \).
- \( LS \) is the hypotenuse we need to find.
Let:
- \( a = YL = 15.4 , \text{km} \)
- \( b = YS = 6.3 , \text{km} \)
- \( c = LS \)
Using the Pythagorean theorem, we have: \[ LS^2 = YL^2 + YS^2 \]
Now, let's calculate \( YL^2 \) and \( YS^2 \): \[ YL^2 = (15.4)^2 = 237.16 \] \[ YS^2 = (6.3)^2 = 39.69 \]
Next, we sum these squares: \[ LS^2 = 237.16 + 39.69 = 276.85 \]
To find \( LS \), we take the square root: \[ LS = \sqrt{276.85} \]
Calculating the square root: \[ LS \approx 16.6268 \]
Now, rounding \( LS \) to the nearest tenth: \[ LS \approx 16.6 , \text{km} \]
Thus, the length of \( LS \) is approximately \( \boxed{16.6} , \text{km} \).