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A parallelogram has side lengths of 4 and 6 and an angle of measure [tex][tex]$55^{\circ}$[/tex][/tex].

Law of cosines: [tex][tex]$a^2 = b^2 + c^2 - 2bc \cos(A)$[/tex][/tex]

What is [tex][tex]$x$[/tex][/tex], the length of the diagonal, to the nearest whole number?

A. 3
B. 5
C. 6
D. 7



Answer :

GinnyAnswer
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]

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