Check solutions to the practice Parallelogram practice problems, exercises, tests:
Area Calculation: A = base × height A = 8 inches × 5 inches = 40 square inches
Perimeter Calculation: P = 2 × (length of base + length of adjacent side) P = 2 × (10 feet + 7 feet) = 34 feet
Diagonal Lengths: d1 = √(base^2 + height^2) d1 = √(6 inches^2 + 4 inches^2) = √(36 + 16) = √52 inches (approximately 7.21 inches)
d2 = √(base^2 + height^2) d2 = √(6 inches^2 + 4 inches^2) = √(36 + 16) = √52 inches (approximately 7.21 inches)
Interior Angles: Opposite angles are congruent, so θ1 = θ2 = 180° - α θ1 = θ2 = 180° - 60° = 120°
Height Calculation: h = A / b h = 72 square inches / 8 inches = 9 inches
Diagonal Length using Law of Cosines: x = √(a^2 + b^2 - 2ab * cos(A)) x = √((9 feet)^2 + (12 feet)^2 - 2(9 feet)(12 feet) * cos(120°)) x = √(81 + 144 - 216 * (-0.5)) x = √(225 + 108) = √333 feet (approximately 18.24 feet)
Area Calculation using Vectors: A = |a × b| A = |(3, 2) × (5, 1)| A = |(31 - 52, 53 - 12)| A = |(-7, 13)| = √((-7)^2 + 13^2) ≈ √218 square units
Sides from Diagonals and Angle: a = √(d1^2 + d2^2 - 2d1d2cos(γ)) a = √((10 inches)^2 + (8 inches)^2 - 2(10 inches)(8 inches) * cos(45°)) a = √(100 + 64 - 160 * 0.7071) a = √(164 - 113.136) ≈ √50.864 inches (approximately 7.12 inches)
b = √(d1^2 + d2^2 + 2d1d2cos(γ)) b = √((10 inches)^2 + (8 inches)^2 + 2(10 inches)(8 inches) * cos(45°)) b = √(100 + 64 + 160 * 0.7071) b = √(164 + 113.136) ≈ √277.136 inches (approximately 16.63 inches)
Sides from Diagonals and Known Side: Using the formula for a: a = √(2d1^2 + 2d2^2 - 4b^2) / 2 a = √(2(6 feet)^2 + 2(8 feet)^2 - 4(4 feet)^2) / 2 a = √(72 + 128 - 64) / 2 a = √(136) / 2 ≈ √34 feet (approximately 5.83 feet)
Sides from Altitude and Angle: Using the formula for a: a = h / sin(α) a = 10 inches / sin(30°) a = 10 inches / 0.5 a = 20 inches
Similarly, b = 20 inches.