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Let's work through this problem together.

### Information provided:
1. \( \overline{RT} \) is the diameter.
2. \( \overline{BE} \) is the radius.
3. \( \overline{RE} \) is a chord.

### Given Data:
- \( m\angle TRE = 15^\circ \)
- \( BE = 9 \)
- We need to find the area of sector \( EBR \).

### Steps to find the area of the sector:
The area of a sector is calculated using the formula:
\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
Where:
- \( \theta \) is the central angle (in degrees), which is \( \angle EBR \) here.
- \( r \) is the radius of the circle, which is \( BE = 9 \).

#### 1. Find the Central Angle \( \angle EBR \):
The problem gives \( m\angle TRE = 15^\circ \). Since \( \overline{RT} \) is the diameter and \( \overline{RE} \) is a chord, the angle \( \angle EBR \) will be double the inscribed angle \( \angle TRE \) due to the properties of circles (the central angle subtended by a chord is twice the inscribed angle).

Thus:
\[
m\angle EBR = 2 \times 15^\circ = 30^\circ
\]

#### 2. Calculate the Area of Sector \( EBR \):
Using the formula for the area of a sector:
\[
\text{Area of sector } EBR = \frac{30^\circ}{360^\circ} \times \pi \times (9)^2
\]

Simplifying:
\[
= \frac{1}{12} \times \pi \times 81 = \frac{81}{12} \pi = 6.75\pi
\]

### Final Answer:
The area of sector \( EBR \) is \( 6.75\pi \). This corresponds to option 2 in the answer choices.

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Would you like more details on any step? Here are some related questions you might find useful:

1. How do you calculate the area of a sector when the angle is given in radians?
2. What are the properties of inscribed and central angles in a circle?
3. How does the diameter of a circle relate to angles formed by chords?
4. What happens to the area of a sector if the angle doubles?
5. How can we determine the length of a chord given the radius and central angle?

**Tip:** When working with sectors in circles, always ensure the angle is in the correct units (degrees or radians) before applying the area formula.

In circle B, diameter RT, radius BE, and chord RE are drawn. If m∠TRE = 15° and BE = 9, what is the area of sector EBR?

This problem demonstrates how to calculate the area of a sector in a circle when the diameter, radius, and inscribed angle are given. Learn how to apply the sector area formula and properties of central and inscribed angles in this detailed step-by-step guide.

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