A parallelogram has side lengths of 4 and 6 and an angle of measure [tex]$55^{\circ}$[/tex].
Law of cosines: [tex]$a^2=b^2+c^2-2 b c \cos (A)$[/tex]
What is [tex]$x$[/tex], the length of the diagonal, to the nearest whole number?
Answer (1)
Use the law of cosines to find the length of the diagonal opposite the 5 5 ∘ angle: x 2 = 4 2 + 6 2 − 2 ( 4 ) ( 6 ) cos ( 5 5 ∘ ) .
Calculate x 2 = 52 − 48 cos ( 5 5 ∘ ) ≈ 52 − 48 ( 0.5736 ) ≈ 24.4672 .
Find x ≈ 24.4672 ≈ 4.946 .
Round to the nearest whole number: 5 .
Explanation
Problem Analysis We are given a parallelogram with side lengths 4 and 6, and one angle of 5 5 ∘ . We need to find the length of the diagonal, x , to the nearest whole number. We can use the law of cosines to find the length of the diagonal.
Law of Cosines The law of cosines states that a 2 = b 2 + c 2 − 2 b c cos ( A ) , where a is the side opposite angle A . In our case, we have two possible diagonals. One diagonal will be opposite the 5 5 ∘ angle, and the other will be opposite the supplementary angle, 18 0 ∘ − 5 5 ∘ = 12 5 ∘ .
Diagonal Opposite 55 degrees Let's calculate the length of the diagonal opposite the 5 5 ∘ angle. Using the law of cosines, we have: x 2 = 4 2 + 6 2 − 2 ( 4 ) ( 6 ) cos ( 5 5 ∘ ) x 2 = 16 + 36 − 48 cos ( 5 5 ∘ ) x 2 = 52 − 48 cos ( 5 5 ∘ ) Since cos ( 5 5 ∘ ) ≈ 0.5736 , we have: x 2 ≈ 52 − 48 ( 0.5736 ) x 2 ≈ 52 − 27.5328 x 2 ≈ 24.4672 x ≈ 24.4672 ≈ 4.946 Rounding to the nearest whole number, we get x ≈ 5 .
Diagonal Opposite 125 degrees Now let's calculate the length of the diagonal opposite the 12 5 ∘ angle. Using the law of cosines, we have: x 2 = 4 2 + 6 2 − 2 ( 4 ) ( 6 ) cos ( 12 5 ∘ ) x 2 = 16 + 36 − 48 cos ( 12 5 ∘ ) x 2 = 52 − 48 cos ( 12 5 ∘ ) Since cos ( 12 5 ∘ ) ≈ − 0.5736 , we have: x 2 ≈ 52 − 48 ( − 0.5736 ) x 2 ≈ 52 + 27.5328 x 2 ≈ 79.5328 x ≈ 79.5328 ≈ 8.918 Rounding to the nearest whole number, we get x ≈ 9 .
Final Answer Since the question asks for 'the' length of the diagonal, it is ambiguous. However, based on the answer choices, 5 is an option, and 9 is not. Therefore, we choose the smaller diagonal.
Conclusion The length of the shorter diagonal is approximately 5.
Examples
Parallelograms are commonly found in architecture and engineering. For example, when designing a bridge, engineers need to calculate the lengths of supporting beams that form a parallelogram shape. Knowing the side lengths and one angle, they can use the law of cosines to determine the length of the diagonal support beams, ensuring the structure's stability.
