We are given the following details about the parallelogram:

• Side $a = 15 \, \text{cm}$
• One diagonal $d_1 = 20 \, \text{cm}$
• The angle between the diagonals $\theta = 40^\circ$
• We need to find the length of the other side $b$.

Using the Law of Cosines

In a parallelogram, the diagonals bisect each other, dividing it into two congruent triangles. The law of cosines can be used to find the length of side $b$.

The diagonals $d_1$ and $d_2$ form a triangle with side lengths $a$ and $b$, and the angle between them is $40^\circ$.

We can use the law of cosines to find $b$: $b^2 = a^2 + d_1^2 - 2 \cdot a \cdot d_1 \cdot \cos(\theta)$

Substituting the values: $b^2 = 15^2 + 20^2 - 2 \cdot 15 \cdot 20 \cdot \cos(40^\circ)$ $b^2 = 225 + 400 - 600 \cdot \cos(40^\circ)$

Using the approximate value $\cos(40^\circ) \approx 0.766$: $b^2 = 225 + 400 - 600 \cdot 0.766$ $b^2 = 625 - 459.6$ $b^2 = 165.4$ $b = \sqrt{165.4} \approx 12.86 \, \text{cm}$

Thus, the length of side $b$ is approximately $12.86 \, \text{cm}$.

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Would you like more details or have any questions about this calculation? Here are 5 related questions to expand the topic:

1. How can we find the area of this parallelogram given the information?
2. What is the formula for the diagonals of a parallelogram in terms of its sides and angles?
3. Can we calculate the length of the second diagonal in this parallelogram?
4. How does the Law of Cosines apply to non-right triangles?
5. How do the properties of diagonals help in determining the type of quadrilateral?

Tip: The Law of Cosines is a powerful tool for solving triangles when you know two sides and the included angle or three sides of a triangle.