In the parallelogram ABCD, AB = 6, AD = 8, AC = 2√13. Find BD.
| May 14, 2021 | education
Since ABCD is a parallelogram, BC = AD = 8 cm.
In the triangle ABC, we apply the cosine theorem and determine the value of the angle ABC.
AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * CosABC.
52 = 36 + 64 – 2 * 6 * 8 * CosABC.
96 * CosABC = 100 – 52 = 48.
CosABC = 48/96 = 1/2.
Angle ABC = arcos (1/2) = 60.
The sum of the adjacent angles of the parallelogram is 180, then the angle ABC = (180 – 60) = 120.
In triangle BAD, by the cosine theorem, we define the length BD
BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * CosBAD.
BD ^ 2 = 36 + 64 – 2 * 6 * 8 * (-1/2) = 100 + 48 = 148.
ВD = 2 * √37 cm.
Answer: The length of the diagonal BD is 2 * √37 cm
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