Question

Знайдіть діагоналі паралелограма, якщо вони відносяться як 3 : 5, а довжини сторін дорівнюють 8 см і 19 см

Answer 1

Пусть одна диагональ $d_1$ равна 3х, а вторая $d_2$ равна 5х.
По теореме косинусов находим длины диагоналей как стороны треугольников.
$d_1^2=8^2+19^2-2*8*19*cosA.$
Так как второй угол В равен 180 - А, то cosB=cos(180-A) = -cosA.
Тогда $d_2^2=8^2+19^2+2*8*19*cosA.$
С учётом того, что $3d_2=5d_1$ составим равенство:
$5(8^2+19^2-2*8*19*cosA)=3(8^2+19^2+2*8*19*cosA).$
$5(425-304cosA)=3(425+304cosa).$
Отсюда находим: 
 $cosA= \frac{2*425}{8*304}= \frac{425}{1216}$ ≈ 0,349507.
Тогда диагональ $d_1=425- \frac{304*425}{1216} =318,75$.
$d_2=425+ \frac{304*425}{1216}=531,25.$

Answer 2

Finding the Diagonals of a Parallelogram

To find the diagonals of a parallelogram, we need to know the ratio between their lengths and the lengths of the sides of the parallelogram. In this case, the ratio between the diagonals is given as 3:5, and the lengths of the sides are given as 8 cm and 19 cm.

Let's denote the lengths of the diagonals as 3x and 5x, where x is a common factor. We can set up the following equation based on the given ratio:

3x / 5x = 8 cm / 19 cm

To solve for x, we can cross-multiply:

3x * 19 cm = 5x * 8 cm

Simplifying the equation:

57x cm = 40x cm

Dividing both sides by x:

57 cm = 40 cm

Since this equation is not true, it means that there is no value of x that satisfies the given ratio and the lengths of the sides. Therefore, we cannot determine the exact lengths of the diagonals of the parallelogram based on the given information.

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