Question

Висоти паралелограма, опущені з його вершини гострого кута, утворюють кут 150 градусив і дорівнюють 5 см і 8 см. Знайдіть стороні паралелограма

Answer 1

Ответ:

10 см, 16 см, абрунтаванне на малюнку

Answer 2

Paraphrased Question:

The heights dropped from the vertices of an acute angle of a parallelogram form an angle of 150 degrees and measure 5 cm and 8 cm. Find the sides of the parallelogram.

Response:

To find the sides of the parallelogram, we can use the properties of parallelograms and trigonometry. Let's denote the sides of the parallelogram as a and b.

Solution:

1. We know that the heights dropped from the vertices of the acute angle form a 150-degree angle. Let's call the height dropped from the vertex opposite side a as h1 and the height dropped from the vertex opposite side b as h2. 2. Using trigonometry, we can determine the relationship between the heights and the sides of the parallelogram. In a right triangle formed by one of the heights and the corresponding side of the parallelogram, the tangent of the acute angle is equal to the ratio of the height to the side length. 3. Applying this to our problem, we can write the following equations: - tan(150 degrees) = h1 / a - tan(150 degrees) = h2 / b 4. We can simplify these equations using the fact that tan(150 degrees) = -tan(30 degrees) = -1/sqrt(3). 5. Simplifying the equations, we get: - h1 / a = -1/sqrt(3) - h2 / b = -1/sqrt(3) 6. Rearranging the equations, we find: - h1 = -a / sqrt(3) - h2 = -b / sqrt(3) 7. We also know the lengths of the heights: h1 = 5 cm and h2 = 8 cm. 8. Substituting these values into the equations, we can solve for a and b: - 5 = -a / sqrt(3) - 8 = -b / sqrt(3) 9. Solving these equations, we find: - a = -5 * sqrt(3) - b = -8 * sqrt(3) 10. Since the sides of a parallelogram cannot be negative, we take the absolute values of a and b: - a = 5 * sqrt(3) - b = 8 * sqrt(3)

Answer:

The sides of the parallelogram are 5 * sqrt(3) cm and 8 * sqrt(3) cm.