To find the length of the diagonal [tex]\( x \)[/tex] in a parallelogram with given side lengths and an angle, we'll use the law of cosines. Here's the step-by-step process:
1. Identify the known values of the parallelogram:
- Side length [tex]\( a = 4 \)[/tex]
- Side length [tex]\( b = 6 \)[/tex]
- Angle [tex]\( A = 55^\circ \)[/tex]
2. Convert the angle [tex]\( A \)[/tex] from degrees to radians:
The angle in degrees needs to be converted to radians because the cosine function in trigonometry typically uses radians.
[tex]\[
A_{radians} = \frac{55 \times \pi}{180} \approx 0.9599 \text{ radians}
\][/tex]
3. Apply the law of cosines:
The law of cosines formula to find the length of a diagonal in a parallelogram is:
[tex]\[
x^2 = a^2 + b^2 - 2ab \cos(A)
\][/tex]
Substituting the known values:
[tex]\[
x^2 = 4^2 + 6^2 - 2 \cdot 4 \cdot 6 \cdot \cos(0.9599)
\][/tex]
4. Calculate the individual components:
[tex]\[
4^2 = 16
\][/tex]
[tex]\[
6^2 = 36
\][/tex]
[tex]\[
2 \cdot 4 \cdot 6 = 48
\][/tex]
[tex]\[
\cos(0.9599) \approx 0.5736
\][/tex]
Putting it all together:
[tex]\[
x^2 = 16 + 36 - 48 \cdot 0.5736
\][/tex]
[tex]\[
x^2 = 16 + 36 - 27.5328
\][/tex]
[tex]\[
x^2 = 52 - 27.5328
\][/tex]
[tex]\[
x^2 \approx 24.4672
\][/tex]
5. Find the square root of [tex]\( x^2 \)[/tex] to get [tex]\( x \)[/tex]:
[tex]\[
x \approx \sqrt{24.4672} \approx 4.9465
\][/tex]
6. Round the result to the nearest whole number:
[tex]\[
x \approx 5
\][/tex]
The length of the diagonal [tex]\( x \)[/tex] to the nearest whole number is:
[tex]\[
\boxed{5}
\][/tex]
