The adjacent sides of parallelogram are 26cm and 28cm and one of its diagonal is 30cm. Find the area of the parallelogram by using heron's formula
Thanks
The answer in my book is 336cm2
You will have 2 congruent triangles, so we need to find only the area of one of them
Heron's formula:
Area = √(s(s-a)(s-b)(s-c)) , were a, b, and c are the sides and s= half the perimeter
s = (26+28+30)/2 = 42
s-a = 16
s-b = 14
s-c = 12
area = √(42x16x14x12) = √112896 = 336
so the area of the whole thing = 672
To find the area of a parallelogram using Heron's formula, we first need to find the lengths of the parallelogram's sides, which we can do using the given information.
In a parallelogram, opposite sides are equal in length. Since the given adjacent sides are 26cm and 28cm, we know that the opposite sides are also of the same length.
Let's denote the length of one of the opposite sides as 'a'. Therefore, the length of the other opposite side would also be 'a'.
We also know that the diagonals of a parallelogram bisect each other. This means that each diagonal will divide the parallelogram into two congruent triangles.
Let's denote the length of the diagonal as 'd'. Therefore, the length of half of the diagonal would be 'd/2'.
Using the given information, we have:
a = 26 cm
d/2 = 30 cm
To find the length of the other opposite side, we can use the Pythagorean theorem. According to the theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, one of the triangles formed by the diagonal and the opposite sides is a right-angled triangle. Using the Pythagorean theorem, we can calculate the length of 'a' as follows:
a^2 = (d/2)^2 + (28 cm)^2
a^2 = (30 cm/2)^2 + (28 cm)^2
a^2 = 15 cm^2 + 784 cm^2
a^2 = 799 cm^2
a = √799 cm (approximately 28.27 cm)
Now that we have the lengths of the opposite sides, we can use Heron's formula to find the area of the parallelogram.
Heron's formula states that the area (A) of a triangle with sides of lengths 'a', 'b', and 'c' is given by:
A = √(s(s - a)(s - b)(s - c))
where 's' is the semi-perimeter of the triangle:
s = (a + b + c) / 2
In our case, since a parallelogram consists of two congruent triangles, we can find the area of one triangle and then double it to get the area of the parallelogram.
Substituting the values into Heron's formula:
s = (26 cm + 28 cm + 30 cm) / 2
s = 27 cm
A = 2 * √(27 cm (27 cm - 26 cm) (27 cm - 28 cm) (27 cm - 30 cm))
A = 2 * √(27 cm * 1 cm * (-1 cm) * (-3 cm))
A = 2 * √(27 cm * 1 cm * 1 cm * 3 cm)
A = 2 * √(81 cm^2)
A = 2 * 9 cm
A = 18 cm^2
Therefore, the area of the parallelogram is 18 square centimeters.