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Interactive Guide
What is a Parallelogram?
A parallelogram is a quadrilateral with opposite sides parallel and equal in length. It has the following properties:
- Opposite sides are parallel and equal in length
- Opposite angles are equal
- Consecutive angles are supplementary (add up to 180°)
- Diagonals bisect each other
Key Formulas
Area:
- A = base × height
- A = ab × sin(α) where a and b are sides and α is the angle between them
Perimeter:
- P = 2(a + b) where a and b are the side lengths
Diagonals:
- d₁ = √(a² + b² + 2ab×cos(α))
- d₂ = √(a² + b² - 2ab×cos(α))
Example Problems
Example 1: A parallelogram has a base of 8 cm and a height of 5 cm. Find its area and perimeter if the other side is 6 cm.
Solution:
Area = 8 × 5 = 40 cm²
Perimeter = 2(8 + 6) = 28 cm
Example 2: A parallelogram has sides of 10 cm and 7 cm with an included angle of 60°. Find its area and diagonals.
Solution:
Area = 10 × 7 × sin(60°) = 10 × 7 × 0.866 = 60.62 cm²
d₁ = √(10² + 7² + 2×10×7×cos(60°)) = 14.18 cm
d₂ = √(10² + 7² - 2×10×7×cos(60°)) = 10.25 cm