To determine the length of the diagonal [tex]\( x \)[/tex] in a parallelogram with side lengths of 4 and 6, and an angle of measure [tex]\( 55^\circ \)[/tex], we can use the Law of Cosines.
The Law of Cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Here, [tex]\( a \)[/tex] will be our diagonal [tex]\( x \)[/tex]. The sides [tex]\( b \)[/tex] and [tex]\( c \)[/tex] of the parallelogram are given as 4 and 6 respectively, and the angle [tex]\( A \)[/tex] is [tex]\( 55^\circ \)[/tex].
We are trying to find:
[tex]\[ x^2 = 4^2 + 6^2 - 2 \cdot 4 \cdot 6 \cdot \cos(55^\circ) \][/tex]
First, let's calculate [tex]\( 4^2 \)[/tex] and [tex]\( 6^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ 6^2 = 36 \][/tex]
Adding these results:
[tex]\[ 16 + 36 = 52 \][/tex]
Next, we need to calculate the product of [tex]\( 2 \cdot 4 \cdot 6 \cdot \cos(55^\circ) \)[/tex]. We know that:
[tex]\[ 2 \cdot 4 \cdot 6 = 48 \][/tex]
Therefore:
[tex]\[ 48 \cos(55^\circ) \][/tex]
Subtract this from 52 to find [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = 52 - 48 \cos(55^\circ) \][/tex]
To find the exact value, we use the fact that [tex]\(\cos(55^\circ)\)[/tex] is approximately 0.5736.
Hence:
[tex]\[ 48 \cdot 0.5736 = 27.5328 \][/tex]
Subtracting this from 52 gives:
[tex]\[ x^2 = 52 - 27.5328 = 24.4672 \][/tex]
Finally, take the square root to find [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{24.4672} \approx 4.946 \][/tex]
Rounding to the nearest whole number, we get:
[tex]\[ x \approx 5 \][/tex]
Therefore, the length of the diagonal [tex]\( x \)[/tex] is given by:
[tex]\[ \boxed{5} \][/tex]
