Question

Высота параллелошрамма апущенные из вершыны тупого угла,равны 10см и 6см.периметр параллелограмма равен 48см.Найдите площадь параллелограмма

Answer 1

Применена формула периметра параллелограмма, формула площади параллелограмма

Answer 2

Given Information

We are given that the heights of the parallelogram dropped from the vertex of the obtuse angle are 10 cm and 6 cm. The perimeter of the parallelogram is 48 cm. We need to find the area of the parallelogram.

Solution

To find the area of the parallelogram, we can use the formula: Area = base × height.

Since the heights of the parallelogram dropped from the vertex of the obtuse angle are given, we need to find the length of the base.

Let's assume the lengths of the sides of the parallelogram are a and b. The perimeter of the parallelogram is given by the formula: Perimeter = 2(a + b).

From the given information, we know that the perimeter of the parallelogram is 48 cm. Therefore, we can write the equation: 48 = 2(a + b).

Simplifying the equation, we get: a + b = 24.

Now, let's consider the triangle formed by the base, height, and one of the sides of the parallelogram. We can use trigonometry to find the length of the base.

In the triangle, the height is the opposite side and the base is the adjacent side to the angle formed by the height and the side of the parallelogram.

Let's assume the angle formed by the height and the side of the parallelogram is θ.

Using the sine function, we can write: sin(θ) = height / side.

Substituting the given values, we have: sin(θ) = 6 / a.

Rearranging the equation, we get: a = 6 / sin(θ).

Similarly, for the other side of the parallelogram, we have: b = 10 / sin(θ).

Now, we can substitute the values of a and b in the equation a + b = 24 to solve for θ.

Simplifying the equation, we get: 6 / sin(θ) + 10 / sin(θ) = 24.

Combining the fractions, we have: (6 + 10) / sin(θ) = 24.

Simplifying further, we get: 16 / sin(θ) = 24.

Cross-multiplying, we have: 16 = 24 sin(θ).

Dividing both sides by 24, we get: sin(θ) = 16 / 24.

Simplifying, we have: sin(θ) = 2 / 3.

Using the inverse sine function, we can find the value of θ.

Calculating the inverse sine of 2/3, we get: θ ≈ 41.81 degrees.

Now that we have the value of θ, we can substitute it back into the equations for a and b to find their values.

Substituting θ into the equation for a, we have: a = 6 / sin(41.81).

Calculating the value of a, we get: a ≈ 9.13 cm.

Substituting θ into the equation for b, we have: b = 10 / sin(41.81).

Calculating the value of b, we get: b ≈ 13.42 cm.

Now that we have the values of a and b, we can calculate the area of the parallelogram using the formula Area = base × height.

Substituting the values, we have: Area = 9.13 cm × 6 cm.

Calculating the area, we get: Area ≈ 54.78 cm².

Therefore, the area of the parallelogram is approximately 54.78 cm².

Answer

The area of the parallelogram is approximately 54.78 cm².