A parallelogram ABCD has the area of 40 cm2, |AB| = 8.5 cm and |BC| = 5.65 cm. Find the length of its diagonals
To find the length of the diagonals of parallelogram ABCD, we need to use the properties of a parallelogram.
First, let's find the height of parallelogram ABCD. The height is the perpendicular distance between any two parallel sides. In this case, we can use the length of side AB as the base and the area of the parallelogram.
Area of a parallelogram = base × height
Substituting the given values:
40 cm^2 = 8.5 cm × height
Solving for the height:
height = 40 cm^2 / 8.5 cm
height ≈ 4.71 cm
Now, the length of one diagonal of a parallelogram can be found using the formula:
diagonal = square root of (base^2 + height^2)
In our case, one diagonal is the length of side AC. Using the given values, we have:
diagonal AC = √(8.5 cm)^2 + (4.71 cm)^2
diagonal AC ≈ √(72.25 cm^2 + 22.14 cm^2)
diagonal AC ≈ √94.39 cm^2
diagonal AC ≈ 9.72 cm
Similarly, the other diagonal of the parallelogram, diagonal BD, has the same length:
diagonal BD = diagonal AC ≈ 9.72 cm
Therefore, the length of both diagonals of parallelogram ABCD is approximately 9.72 cm.
If the altitude from D to AB has length h, and meets AB at a distance x from A, then
8.5h = 40
h = 40/8.5 = 4.7
x^2 + h^2 = 5.65^2
x = 3.13
tanA = h/x = 1.5
A = 56.4°
Now use the law of cosines to find the diagonals.