the altitude BE of parallelogram ABCD divides the side AD into segments in the ratio 1:3. Find the area of the parallelogram if the length of its shorter side is 14 cm, and one of its interior angle measures 60 degrees.
196 sqrt3 cm*2
To find the area of the parallelogram, we first need to find the length of the altitude BE.
Let's denote the length of AD as x.
According to the given information, the segment AD is divided into segments in the ratio 1:3. This means that the length of AE is x/4 and the length of DE is (3/4)x.
Now, let's find the length of the altitude BE:
BE = √((AD)^2 - (AE)^2) [using Pythagorean theorem]
BE = √(x^2 - (x/4)^2)
BE = √(16/16)x^2 - (1/16)x^2)
BE = √(15/16)x^2
BE = (x√15)/4
Given that the length of the shorter side AB is 14 cm, we can use the formula for the area of a parallelogram:
Area of parallelogram = base x height
Since the base AB has length 14 cm and the height BE has length (x√15)/4, we can substitute these values into the formula:
Area of parallelogram = 14 cm x (x√15)/4
Next, we need to determine the value of x. To do this, we can use the fact that one of the interior angles of the parallelogram ABCD measures 60 degrees.
In a parallelogram, opposite angles are equal. Since one of the interior angles measures 60 degrees, the opposite angle is also 60 degrees. Therefore, the other interior angles are 180 - 60 = 120 degrees each.
Since both opposite angles are equal in a parallelogram, the opposite angle to the one measuring 120 degrees is also 120 degrees.
We can use the Law of Cosines to find the length of AD:
(AD)^2 = (AE)^2 + (DE)^2 - 2(AE)(DE)cos(120)
(x)^2 = (x/4)^2 + (3/4)x^2 - 2(x/4)(3/4)x(-1/2)
x^2 = (1/16)x^2 + (9/16)x^2 + (3/8)x^2
16x^2 = x^2 + 9x^2 + 6x^2
16x^2 = 16x^2
Therefore, the equation is satisfied for any value of x.
Now let's substitute the value of x into the formula for the area of the parallelogram:
Area of parallelogram = 14 cm x ((x√15)/4)
Area of parallelogram = (14x√15) cm^2
So, the area of the parallelogram ABCD is 14x√15 square centimeters.
To find the area of the parallelogram, we need to find the length of the altitude BE and the length of the longer side AD.
Let's start by finding the length of the shorter side AD. We are given that its length is 14 cm.
Next, we need to find the length of the altitude BE and the longer side AD. We are told that the altitude divides AD into segments in the ratio 1:3.
Since the shorter side AD is divided into segments in the ratio 1:3, we can set up the following proportion:
AD / BE = 3 / 1
We know that the length of the shorter side AD is 14 cm, so we can substitute this into the proportion:
14 / BE = 3 / 1
Cross-multiplying, we get:
14 = 3BE
Dividing both sides by 3, we find:
BE = 14 / 3
Now, to find the length of the longer side AD, we add the lengths of the segments created by the altitude to the length of BE:
AD = BE (shorter segment) + BE (longer segment)
AD = (14 / 3) + (14 / 3) = (2 * 14) / 3 = 28 / 3
Next, we need to find the length of the diagonals of the parallelogram. Since the diagonals of a parallelogram bisect each other, we can find the length of one of the diagonals by dividing the length of AD by 2:
Diagonal = AD / 2 = (28 / 3) / 2 = 14 / 3
We can now find the length of the longer side, which is equal to 2 times the length of the diagonal. So:
AD = 2 * Diagonal = 2 * (14 / 3) = 28 / 3
Now, we have the lengths of the sides of the parallelogram. We can find the area by multiplying the length of the shorter side by the altitude BE:
Area = shorter side * altitude = 14 * (14 / 3) = 196 / 3 cm²
Therefore, the area of the parallelogram is 196 / 3 cm².