In a parallelogram p q r s shown below pq is 8 cm and angle spq is 30° if the area of parallelogram is 24 cm squared find the perimeter?

To find the perimeter of the parallelogram, we need to find the lengths of the sides.

Given that the area of the parallelogram is 24 cm squared, we can use the formula for the area of a parallelogram:

Area = base x height

Since the base is PQ and the height is the perpendicular distance from point S to line PQ, we can find the height using trigonometry.

In triangle SPQ, we have the angle SPQ = 30°, PQ = 8 cm, and we want to find the height PS.

Using trigonometry, we can write:

sin(30°) = PS / PQ
sin(30°) = PS / 8
PS = 8 * sin(30°)
PS = 8 * 0.5
PS = 4 cm

Now that we have the height PS, we can find the length of the side SR by using the Pythagorean theorem in triangle PSR:

SR^2 = PS^2 + PQ^2
SR^2 = 4^2 + 8^2
SR^2 = 16 + 64
SR^2 = 80
SR = √80
SR ≈ 8.94 cm

Since the opposite sides of a parallelogram are equal in length, the length of side QR is also 8.94 cm.

Now we can find the perimeter of the parallelogram:

Perimeter = 2(PQ + QR)
Perimeter = 2(8 + 8.94)
Perimeter = 2(16.94)
Perimeter = 33.88 cm

Therefore, the perimeter of the parallelogram is 33.88 cm.

Step 1: Find the height (SR)

In a parallelogram, the height (SR) is the perpendicular distance between the bases (PQ and RS). We can use trigonometry to find SR:

```
sin(30°) = SR / PQ
```

```
SR = PQ - sin(30°)
SR = 8 cm - sin(30°)
SR = 4 cm
```

Step 2: Find the length of RS

The area of a parallelogram is given by:

```
Area = base - height
```

```
24 cm² = PQ - SR
24 cm² = 8 cm - 4 cm
```

```
RS = 3 cm
```

Step 3: Find the perimeter

The perimeter of a parallelogram is the sum of the lengths of all four sides:

```
Perimeter = PQ + QR + RS + SP
```

Since PQ and RS are opposite sides, they are equal:

```
Perimeter = 2 - PQ + 2 - RS
```

```
Perimeter = 2 - 8 cm + 2 - 3 cm
Perimeter = 16 cm + 6 cm
Perimeter = 22 cm
```

Therefore, the perimeter of the parallelogram is 22 cm.