Determine the area of a rectangle whose diagonal is 24cm and the angle between  the diagonals is 60°.

We are given:

- The length of the diagonal \( d = 24 \, \text{cm} \)
- The angle between the diagonals \( \theta = 60^\circ \)

We can determine the area of the rectangle using the following method:

### Step 1: Relationship between diagonals and sides
Let the sides of the rectangle be \( a \) and \( b \). The diagonals of a rectangle are equal and intersect at 90°. The diagonals split the rectangle into two congruent right triangles. Using the Law of Cosines on one of these triangles, where the angle between the diagonals is \( 60^\circ \), the formula for the diagonal can be derived:

\[
d^2 = a^2 + b^2
\]

Additionally, the area of the rectangle can be related to the product of its diagonals and the sine of the angle between them.

### Step 2: Formula for area in terms of diagonals and angle between them
The area \( A \) of a rectangle is also given by the formula:

\[
A = \frac{1}{2} d^2 \sin(\theta)
\]

### Step 3: Substitute the given values
Substitute \( d = 24 \, \text{cm} \) and \( \theta = 60^\circ \) into the formula:

\[
A = \frac{1}{2} \times 24^2 \times \sin(60^\circ)
\]
\[
A = \frac{1}{2} \times 576 \times \frac{\sqrt{3}}{2}
\]
\[
A = 288 \times \frac{\sqrt{3}}{2}
\]
\[
A = 144\sqrt{3} \, \text{cm}^2
\]

Thus, the area of the rectangle is \( 144\sqrt{3} \, \text{cm}^2 \), which is approximately \( 249.4 \, \text{cm}^2 \).

Would you like more details on this solution, or do you have any questions?

### Related questions:
1. How do you derive the Law of Cosines in this context?
2. What is the relationship between the diagonals and sides of a rectangle?
3. How is the sine function used to calculate areas involving angles?
4. Can the area formula be extended to other quadrilaterals?
5. How does the shape of the rectangle change when the angle between diagonals is different?

### Tip:
Remember, for any parallelogram (including rectangles), the area can be found using diagonals and the sine of the angle between them!

Determine the area of a rectangle whose diagonal is 24cm and the angle between the diagonals is 60°.

Learn how to calculate the area of a rectangle with a diagonal of 24 cm and an angle of 60° between the diagonals. This problem utilizes trigonometric concepts and the Law of Cosines, making it suitable for students in Grades 9-12.

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