In a triangle PQR, let `anglePQR = 30^(@)` and the sides PQ and QR have lengths `10 sqrt3` and 10, respectively. Then, which of the following statement(s) is (are) TRUE ?

Options

A. `angleQPR = 45^(@)`
B. The area of the triangle PQR is `25sqrt3 and angleQRP = 120^(@)`
C. The radius of the incircle of the triangle PQR is `10 sqrt3 -15`
D. The area of the circumcircle of the triangle PQR is `100 pi`

Explanation

To solve the problem, we will analyze triangle PQR with the given information. We know that:

- \( \angle PQR = 30^\circ \)
- \( PQ = 10\sqrt{3} \)
- \( QR = 10 \)

We need to find the length of side PR and then evaluate the truth of the given statements.

### Step 1: Use the Law of Cosines

We can use the Law of Cosines to find the length of side PR. The Law of Cosines states:

\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Where:
- \( c \) is the side opposite angle \( C \) (in this case, PR),
- \( a \) and \( b \) are the other two sides (PQ and QR),
- \( C \) is the angle between sides \( a \) and \( b \).

In our case:
- \( a = PQ = 10\sqrt{3} \)
- \( b = QR = 10 \)
- \( C = \angle PQR = 30^\circ \)

Substituting these values into the Law of Cosines:

\[ PR^2 = (10\sqrt{3})^2 + (10)^2 - 2 \cdot (10\sqrt{3}) \cdot (10) \cdot \cos(30^\circ) \]

### Step 2: Calculate the Values

Calculating each term:

\[ (10\sqrt{3})^2 = 100 \cdot 3 = 300 \]

\[ (10)^2 = 100 \]

\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

Now substituting these into the equation:

\[ PR^2 = 300 + 100 - 2 \cdot (10\sqrt{3}) \cdot (10) \cdot \frac{\sqrt{3}}{2} \]

Calculating the last term:

\[ 2 \cdot (10\sqrt{3}) \cdot (10) \cdot \frac{\sqrt{3}}{2} = 100 \cdot 3 = 300 \]

Now substituting this back into the equation:

\[ PR^2 = 300 + 100 - 300 \]

\[ PR^2 = 100 \]

Taking the square root:

\[ PR = 10 \]

### Step 3: Evaluate the Statements

Now that we have \( PR = 10 \), we can evaluate the statements provided in the options.

1. **Statement A**: \( \angle PQR = 45^\circ \) - **False**, as \( \angle PQR = 30^\circ \).
2. **Statement B**: Area of triangle PQR is 120 - **False**, as we will calculate the area next.
3. **Statement C**: Area of triangle PQR is \( 25\sqrt{3} \) - **True**.
4. **Statement D**: Circumradius \( R \) is 10 - **True**.

### Step 4: Calculate the Area of Triangle PQR

The area of triangle PQR can be calculated using the formula:

\[ \text{Area} = \frac{1}{2} \times PQ \times QR \times \sin(\angle PQR) \]

Substituting the known values:

\[ \text{Area} = \frac{1}{2} \times (10\sqrt{3}) \times (10) \times \sin(30^\circ) \]

Since \( \sin(30^\circ) = \frac{1}{2} \):

\[ \text{Area} = \frac{1}{2} \times (10\sqrt{3}) \times (10) \times \frac{1}{2} \]

\[ = \frac{1}{4} \times 100\sqrt{3} = 25\sqrt{3} \]

### Conclusion

The true statements are:
- Statement C: Area of triangle PQR is \( 25\sqrt{3} \).
- Statement D: Circumradius \( R \) is 10.