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To solve this problem, we need to find the area of the shaded region, which is the area of the circle minus the area of the inscribed right triangle \(TUV\).

### Step 1: Identify the radius of the circle
The right triangle \(TUV\) is inscribed in the circle, meaning that the hypotenuse \(UV\) of the triangle is the diameter of the circle. We are given the two legs of the triangle:
- \(TU = 2.4\)
- \(TV = 1.8\)

Using the Pythagorean Theorem to find the hypotenuse \(UV\):
\[
UV = \sqrt{TU^2 + TV^2} = \sqrt{2.4^2 + 1.8^2}
\]
\[
UV = \sqrt{5.76 + 3.24} = \sqrt{9} = 3
\]
The diameter \(UV\) of the circle is 3, so the radius \(r\) is:
\[
r = \frac{3}{2} = 1.5
\]

### Step 2: Calculate the area of the circle
The area \(A\) of the circle is given by:
\[
A = \pi r^2 = \pi (1.5)^2 = \pi \times 2.25 \approx 7.07 \, \text{square units}
\]

### Step 3: Calculate the area of the triangle \(TUV\)
The area \(A_{\text{triangle}}\) of the triangle can be calculated using the formula for the area of a right triangle:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2.4 \times 1.8 = \frac{1}{2} \times 4.32 = 2.16 \, \text{square units}
\]

### Step 4: Find the area of the shaded region
The area of the shaded region is the area of the circle minus the area of the triangle:
\[
A_{\text{shaded}} = A - A_{\text{triangle}} \approx 7.07 - 2.16 = 4.91 \, \text{square units}
\]

### Final Answer
The area of the shaded region is approximately **4.9 square units** (rounded to the nearest tenth).

Would you like more details or have any other questions?

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Here are five related questions you might consider:

1. How would the solution change if the triangle was not a right triangle?
2. What is the relationship between the radius of the circle and the hypotenuse of an inscribed right triangle?
3. How can you determine the area of a segment of a circle instead of the shaded region?
4. How would you solve this problem if only the area of the triangle was given, but not the side lengths?
5. How does changing one side of the triangle affect the area of the shaded region?

**Tip:** When solving geometry problems involving circles and inscribed shapes, always remember to consider relationships like the Pythagorean theorem and properties of tangents and secants.

Right triangle TUV is inscribed in circle X. Find the area of the shaded region. Round your answer to the nearest tenth if necessary.

Learn how to find the area of a shaded region formed by a right triangle inscribed in a circle. This problem involves applying the Pythagorean theorem to calculate the hypotenuse, determining the radius of the circle, and subtracting the area of the triangle from the area of the circle. Suitable for students in Grades 7-9.

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