Question

For context, there's a right triangle with top being L, bottom right S, and bottom left Y.

If YS=6.3 km and YL=15.4 km, find the length of LS. Round your answer to the nearest tenth if necessary. You must show ALL work.

Answer 1

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 \( \boxed{16.6} \, \text{km} \).