To find angle \( x \) using the Law of Cosines, we can apply the formula:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(x) \]
where:
- \( c \) is the length of the hypotenuse,
- \( a \) and \( b \) are the lengths of the other two sides.
In your triangle:
- \( c = 9.2 \) (the hypotenuse),
- \( a = 5.0 \) (one leg),
- \( b = 7.7 \) (the other leg).
Plugging in the values:
\[ (9.2)^2 = (5.0)^2 + (7.7)^2 - 2 \cdot (5.0) \cdot (7.7) \cdot \cos(x) \]
Calculating the squares:
\[ 84.64 = 25 + 59.29 - 77 \cdot \cos(x) \]
Now combine the known values:
\[ 84.64 = 84.29 - 77 \cdot \cos(x) \]
Subtract 84.29 from both sides:
\[ 84.64 - 84.29 = -77 \cdot \cos(x) \]
\[ 0.35 = -77 \cdot \cos(x) \]
Now, divide both sides by -77 to isolate \( \cos(x) \):
\[ \cos(x) = \frac{-0.35}{77} \]
Calculating the right side:
\[ \cos(x) \approx -0.004545 \]
To find \( x \), take the arccosine:
\[ x = \cos^{-1}(-0.004545) \]
Using a calculator:
\[ x \approx 90.3° \]
Thus, rounding to the nearest tenth, we find:
x is equal to 90.3 degrees.
So the correct response is:
x=90.3° x is equal to 90.3 degrees