Question

For context, there's a right triangle with top point L, bottome right point S, bottom left point 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.
Show ALL work.

Answer 1

To find the length of \( LS \) in the right triangle \( LYS \) with the given points, we can use the Pythagorean theorem. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.

In triangle \( LYS \):
- The length \( YS \) (one leg) is given as \( 6.3 \) km.
- The length \( YL \) (the other leg) is given as \( 15.4 \) km.
- The length \( LS \) will be the hypotenuse of the triangle.

Using the Pythagorean theorem:
\[
YL^2 + YS^2 = LS^2
\]

Substituting the given lengths:
\[
(15.4)^2 + (6.3)^2 = LS^2
\]

Calculating each term:
\[
15.4^2 = 237.16
\]
\[
6.3^2 = 39.69
\]

Now add these results together:
\[
237.16 + 39.69 = LS^2
\]
\[
LS^2 = 276.85
\]

Next, we take the square root to find \( LS \):
\[
LS = \sqrt{276.85} \approx 16.628
\]

Rounding to the nearest tenth:
\[
LS \approx 16.6 \text{ km}
\]

Thus, the length of \( LS \) is approximately **16.6 km**.