To find the length of the side opposite the included angle in the oblique triangle, we can use the Law of Cosines. The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
where:
- \( c \) is the side opposite angle \( C \),
- \( a \) and \( b \) are the lengths of the other two sides,
- \( C \) is the included angle.
In this case:
- \( a = 17 , \text{cm} \)
- \( b = 28 , \text{cm} \)
- \( C = 91^\circ \)
We can plug these values into the formula to find \( c \):
\[ c^2 = 17^2 + 28^2 - 2 \cdot 17 \cdot 28 \cdot \cos(91^\circ) \]
Calculating \( 17^2 \) and \( 28^2 \):
\[ 17^2 = 289 \] \[ 28^2 = 784 \]
Now substitute these into the equation:
\[ c^2 = 289 + 784 - 2 \cdot 17 \cdot 28 \cdot \cos(91^\circ) \]
Next, we compute \( \cos(91^\circ) \). Since \( 91^\circ \) is very close to \( 90^\circ \), it is slightly negative. Using a calculator, we find:
\[ \cos(91^\circ) \approx -0.015 \]
Now substituting this value into the equation:
\[ c^2 = 289 + 784 - 2 \cdot 17 \cdot 28 \cdot (-0.015) \]
Calculating \( 2 \cdot 17 \cdot 28 \):
\[ 2 \cdot 17 \cdot 28 = 952 \]
Now, substituting back:
\[ c^2 = 289 + 784 + 952 \cdot 0.015 \] \[ c^2 = 289 + 784 + 14.28 \] \[ c^2 = 1073.28 \]
Now, take the square root to find \( c \):
\[ c = \sqrt{1073.28} \approx 32.8 \]
Rounding to the nearest whole number gives us:
\[ c \approx 33 , \text{cm} \]
Therefore, the measure of the side opposite the 91-degree angle is approximately:
\[ \boxed{33} , \text{cm} \]
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