To find the length of side PQ in triangle PQR, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides \( a \), \( b \), and \( c \), and an angle \( C \) opposite side \( c \):
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
In triangle PQR:
- Let \( a = PR = 7 , \text{cm} \)
- Let \( b = RQ = 9 , \text{cm} \)
- Let \( C = \angle PQR = 108^\circ \)
- Let \( c = PQ \) (which we want to find)
Using the Law of Cosines:
\[ PQ^2 = PR^2 + RQ^2 - 2 \cdot PR \cdot RQ \cdot \cos(\angle PQR) \]
Substituting the values:
\[ PQ^2 = 7^2 + 9^2 - 2 \cdot 7 \cdot 9 \cdot \cos(108^\circ) \]
Calculating each term:
- \( 7^2 = 49 \)
- \( 9^2 = 81 \)
- \( PR \cdot RQ = 7 \cdot 9 = 63 \)
Now we need to find \( \cos(108^\circ) \). From trigonometric tables or a calculator, we know:
\[ \cos(108^\circ) \approx -0.309 \]
Now plug in this value:
\[ PQ^2 = 49 + 81 - 2 \cdot 63 \cdot (-0.309) \]
Calculating:
- \( 49 + 81 = 130 \)
- \( -2 \cdot 63 \cdot (-0.309) = 2 \cdot 63 \cdot 0.309 \approx 39.078 \)
Now, add these results:
\[ PQ^2 \approx 130 + 39.078 = 169.078 \]
To find PQ, take the square root:
\[ PQ \approx \sqrt{169.078} \approx 13.0 , \text{cm} \]
Thus, the length of side PQ is approximately 13.0 cm.