The diagonal of a parallelogram 6cm and 8cm long and
they intersect at an angle of 55 degrees. calculate the area of
the parallelogram
The diagonals bisect each other, and also meet at an angle of 125°. So, the long side of the parallelogram can be found using the law of cosines.
Let the parallelogram be ABCD and the center be E.
AB^2 = 3^2+4^2 - 2*3*4*cos125°
AB = 6.2262
Now, the altitude of triangle AEB is found using the law of sines:
sin(EAB)/3 = sin(125°)/6.2262
EAB = 23.25°
h/4 = sin 23.25°
h = 1.578
So, the area of the parallelogram is
1/2 * 6.2262 * (2*1.578) = 9.825
Or, using vectors, let
u = 4i
v = 3cos55° i + 3sin55° j
Then the area is
1/2 |u×v| = 9.829
close enough
To calculate the area of a parallelogram, we need to know the length of the base and the height. In this case, since we know the lengths of the diagonals, we can use them to find the base and height.
Let's start by finding the length of the base:
Since the diagonals of a parallelogram bisect each other, we can use the properties of right triangles to find the length of the base.
Using the Pythagorean theorem for the right triangle formed by the diagonal of lengths 6 cm, 8 cm, and the base:
Base^2 + Height^2 = Diagonal^2
Base^2 + Height^2 = 6^2
Base^2 + Height^2 = 36
Now, let's find the length of the base.
Since the diagonals intersect at an angle of 55 degrees, we can use the trigonometric functions to find the height.
tan(55) = Height/Base
Height = Base * tan(55)
Now, we have two equations that we can solve simultaneously to find the base and height.
Base^2 + (Base*tan(55))^2 = 36
Expanding the equation:
Base^2 + Base^2 * tan^2(55) = 36
Now, let's solve this equation to find the value of the base.
2 * Base^2 * tan^2(55) = 36
Base^2 * tan^2(55) = 18
Base^2 = 18 / tan^2(55)
Base = sqrt(18 / tan^2(55))
Calculating this using a calculator, we find:
Base ≈ 4.592 cm
Now that we have the base, we can use it to find the height:
Height = Base * tan(55)
Height = 4.592 * tan(55)
Calculating this using a calculator, we find:
Height ≈ 4.948 cm
Finally, we can calculate the area of the parallelogram using the formula:
Area = Base * Height
Area ≈ 4.592 * 4.948
Calculating this using a calculator, we find:
Area ≈ 22.707 cm^2
Therefore, the area of the parallelogram is approximately 22.707 square centimeters.
To calculate the area of a parallelogram, you need the length of the base and the height. In this case, we have the lengths of the diagonals instead. However, we can find the base and height using the given information.
Step 1: Finding the base
The diagonals of a parallelogram bisect each other. This means that they divide each other into two equal parts. Since we are given the lengths of both diagonals, we can find the length of the base by using the formula for the bisected diagonal, which is:
d1^2 = b^2 + h^2
where d1 is the longer diagonal, b is the length of the base, and h is the length of the height.
In this case, given:
d1 = 8 cm
Let's solve for b:
8^2 = b^2 + h^2
64 = b^2 + h^2
Step 2: Finding the height
To find the height, we need to know the angle at which the diagonals intersect.
Given:
Angle = 55 degrees
In a parallelogram, the diagonals intersect each other at opposite angles. So, the opposite angle will also be 55 degrees.
We can use the sine rule to find the height (h):
sin(angle) = h / d2
where d2 is the shorter diagonal.
In this case, given:
Angle = 55 degrees
d2 = 6 cm
Let's solve for h:
sin(55) = h / 6
h = 6 * sin(55)
Step 3: Calculating the area
Now that we have the length of the base (b) and the height (h), we can calculate the area of the parallelogram using the formula:
Area = base * height
In this case:
Area = b * h
Now you can substitute the values we found into the formula and calculate the area.