In parallelogram ABCD, ∠BAD=60∘, AB=6, and diagonal AC=631. Find the area of ABCD.
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Step-by-Step Solution
Step 1: Convert mixed fraction to improper fraction
First, we convert the mixed fraction for the length of diagonal AC into an improper fraction. This makes calculations easier and more precise.
Step 2: Apply the Law of Cosines in triangle ABC
In a parallelogram, consecutive angles are supplementary. So, ∠ABC=180∘−∠BAD=180∘−60∘=120∘. We can use the Law of Cosines in △ABC to find the length of side BC. The Law of Cosines states that c2=a2+b2−2abcos(C).
Step 3: Substitute values and solve for BC
Substitute the known values into the Law of Cosines equation. We know AC=319, AB=6, and cos(120∘)=−21. This results in a quadratic equation for BC.
Step 4: Solve the quadratic equation for BC
Multiply the quadratic equation by 9 to clear the fraction. Then, use the quadratic formula x=2a−b±b2−4ac to solve for BC. Since BC must be a positive length, we take the positive root.
Step 5: Determine the valid length of BC
Since length cannot be negative, we choose the positive value for BC. The square root of 118 is approximately 10.86, so 3−9+10.86>0.
Step 6: Calculate the area of the parallelogram
The area of a parallelogram is given by the formula A=absin(θ), where a and b are the lengths of two adjacent sides and θ is the angle between them. We use AB=6, BC=3−9+118, and ∠BAD=60∘.