Tick the correct answer and justify : In `DeltaPQR `, PQ = `6sqrt(3)` cm , PR =12 cm and QR = 6 cm. The angle Q is :

Options

A. `120^(@)`
B. `60^(@)`
C. `90^(@) `
D. `45^(@)`

Answer

To solve for the angle Q in triangle PQR, where PQ = \(6\sqrt{3}\) cm, PR = 12 cm, and QR = 6 cm, we can use the Pythagorean theorem. Here are the steps:

### Step 1: Identify the sides of the triangle
We have:
- PQ = \(6\sqrt{3}\) cm
- PR = 12 cm
- QR = 6 cm

### Step 2: Square the lengths of the sides
Calculate the squares of each side:
- \(PQ^2 = (6\sqrt{3})^2 = 6^2 \times (\sqrt{3})^2 = 36 \times 3 = 108\)
- \(QR^2 = 6^2 = 36\)
- \(PR^2 = 12^2 = 144\)

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse (PR) is equal to the sum of the squares of the other two sides (PQ and QR):
\[ PQ^2 + QR^2 = PR^2 \]
Substituting the values we calculated:
\[ 108 + 36 = 144 \]
This simplifies to:
\[ 144 = 144 \]
This confirms that triangle PQR is a right triangle.

### Step 4: Determine the angle Q
Since we have established that triangle PQR is a right triangle with PR as the hypotenuse, the angle opposite to the hypotenuse (which is angle Q) must be 90 degrees.

### Final Answer
Thus, the angle Q is \(90^\circ\).