To solve for the area of the rectangle, we analyze the given problem step by step:

Step 1: Recognize the 30°-60°-90° triangle properties

In a 30°-60°-90° triangle, the relationships between the sides are as follows:

1. The hypotenuse is twice the shorter leg.
2. The longer leg is $\sqrt{3}$ times the shorter leg.

We are given:

• The diagonal of the rectangle (hypotenuse of the triangle) is 10.
• The diagonal divides the rectangle into a 30°-60°-90° triangle.

Let:

• $x$ = shorter leg (opposite $30^\circ$),
• $x\sqrt{3}$ = longer leg (opposite $60^\circ$).

Step 2: Find the shorter leg

The hypotenuse is twice the shorter leg, so:

$2x = 10 \implies x = \frac{10}{2} = 5$

Thus, the shorter leg $x$ is 5.

Step 3: Find the longer leg

The longer leg is given by:

$x\sqrt{3} = 5\sqrt{3}$

Step 4: Calculate the area of the rectangle

The rectangle's area is:

$\text{Area} = \text{length} \times \text{width}$

Here:

• Length = $5\sqrt{3}$ (longer leg),
• Width = $5$ (shorter leg).

$\text{Area} = 5 \times 5\sqrt{3} = 25\sqrt{3}$

Step 5: Final Answer

The area of the rectangle is:

B. $25\sqrt{3}$

Do you need further explanation or another example?

Here are 5 related questions to deepen understanding:

1. How do the properties of a 30°-60°-90° triangle help in solving geometric problems?
2. How can you derive the area of a rectangle using diagonals and angles?
3. What are the side ratios of a right triangle with angles 45°-45°-90°?
4. How do trigonometric functions relate to special right triangles?
5. How would the solution change if the diagonal were a different length?

Tip: Always remember the side ratios of special right triangles (30°-60°-90° and 45°-45°-90°) as they simplify calculations significantly.