In a parallelogram abcd, ab = 4, ac = 5, bc = 3. Find the area of a parallelogram.

univerkov | January 7, 2021 | education

Since the opposite sides of the parallelogram are equal, then СD = AB = 4 cm.
The diagonal AC of the parallelogram divides it into two triangles, ABC and ADС in which three sides are equal, and therefore the triangles are equal.
In the AСD triangle, according to Heron’s theorem, we define its area.
The semi-perimeter of the AСD triangle is equal to: p = (AС + СD + AD) / 2 = (5 + 4 + 3) / 2 = 12/2 = 6 cm.
Then Sacd = √р * (р – СD) * (р – AD) * (р – АС) = √6 * (6 – 4) * (6 – 3) * (6 – 5) = √6 * 2 * 3 * 1 = √3 = 6 cm2.
Then Savsd = 2 * Sasd = 2 * 6 = 12 cm2.
Answer: The area of the parallelogram is 12 cm2.

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