https://file.aiquickdraw.com/mathbot/file/1726363452253rj727cwt.jpg

We are given that diameters \( IJ = JK = KL \), and \( IO = 12 \), and we are tasked with finding the area of the shaded region in the figure.

### Step-by-step approach:

1. **Given Information**:
   - \( IJ = JK = KL \) implies that each of these segments represents one-third of the entire diameter \( IL \).
   - \( IO = 12 \) is the radius of the entire circle.

2. **Diameter of the circle**:
   Since \( IO = 12 \), the diameter \( IL \) of the entire circle is \( 2 \times 12 = 24 \).

3. **Dividing the circle**:
   - The problem specifies that \( IJ = JK = KL \), meaning the diameter is divided into three equal parts.
   - Thus, each part (like \( IJ \)) is \( \frac{24}{3} = 8 \).

4. **Areas of the sectors**:
   - The entire circle has an area given by the formula \( A = \pi r^2 \), where \( r = 12 \). Thus, the area of the full circle is:
   \[
   A_{\text{full}} = \pi \times 12^2 = 144\pi
   \]

   - The two shaded regions combined form a semicircle. Since these two sectors together form half the full circle, the area of the semicircle is:
   \[
   A_{\text{semicircle}} = \frac{1}{2} \times 144\pi = 72\pi
   \]

5. **Unshaded area**:
   - The unshaded region is the central circular segment formed by \( JK \), which has a radius of 8 (since \( JK \) is part of the diameter and equals \( 8 \)). The area of this circle is:
   \[
   A_{\text{middle}} = \pi \times 8^2 = 64\pi
   \]

6. **Shaded area**:
   - The shaded area is the difference between the area of the semicircle and the middle circle:
   \[
   A_{\text{shaded}} = 72\pi - 64\pi = 8\pi
   \]

Thus, the area of the shaded region is \( \boxed{8\pi} \).

Let me know if you'd like further details or explanations.

### Related Questions:
1. How do you calculate the area of a sector in general?
2. What is the formula for the area of a full circle, and how does it apply to this problem?
3. How can we find the area of a semicircle?
4. If \( IO \) was 15 instead of 12, how would that change the final answer?
5. What if the unshaded region was a quarter of the circle instead of a semicircle?

**Tip:** Always check the relationships between different parts of a diagram, such as how diameters divide or segment a shape, to correctly apply area formulas.

If diameters IJ = JK = KL and IO = 12, what is the area of the shaded region?

Solve a geometry problem involving a circle where diameters IJ, JK, and KL are equal, and IO = 12. Learn how to calculate the area of the shaded region using circle area formulas. This problem is suitable for students in Grades 8-10.

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