To determine the length of the diagonal [tex]\( x \)[/tex] of a parallelogram with given side lengths and an angle, we will use the Law of Cosines. Here is a step-by-step solution:
1. Identify the given values:
- Side lengths: [tex]\( a = 13 \)[/tex] and [tex]\( b = 17 \)[/tex]
- Included angle: [tex]\( \angle A = 84^\circ \)[/tex]
2. Convert the angle from degrees to radians:
- The angle in radians is approximately [tex]\( 1.466 \, \text{radians} \)[/tex].
3. Apply the Law of Cosines to find the length of the diagonal:
The Law of Cosines formula for the length of the diagonal (let's suppose [tex]\( c \)[/tex] as the diagonal length) is:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(\angle A)
\][/tex]
Substituting the given values:
[tex]\[
c^2 = 13^2 + 17^2 - 2 \cdot 13 \cdot 17 \cdot \cos(84^\circ)
\][/tex]
4. Calculate the value of [tex]\( c^2 \)[/tex]:
- Calculation results in [tex]\( c^2 \approx 411.798 \)[/tex]
5. Find the length of [tex]\( c \)[/tex] by taking the square root:
[tex]\[
c = \sqrt{411.798} \approx 20.293
\][/tex]
6. Round the length of the diagonal to the nearest whole number:
- The rounded value is [tex]\( 20 \)[/tex].
So the length of the diagonal [tex]\( x \)[/tex] to the nearest whole number is [tex]\( \boxed{20} \)[/tex].
