In `anglePQR`, if PQ = 6 cm , PR = 8 cm, QS = 3 cm and PS is the bisector of `angle QPR`, what is the length of QR ?

Answer

To find the length of QR in triangle PQR given that PS is the angle bisector of angle QPR, we can use the Angle Bisector Theorem. Here’s the step-by-step solution:

### Step 1: Understand the given information
We are given:
- PQ = 6 cm
- PR = 8 cm
- QS = 3 cm
- PS is the angle bisector of angle QPR

### Step 2: Apply the Angle Bisector Theorem
According to the Angle Bisector Theorem, the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the other two sides of the triangle. Therefore, we can write:

\[ \frac{PQ}{QR} = \frac{PS}{SR} \]

Where:
- PQ = 6 cm
- PR = 8 cm
- QS = 3 cm
- Let SR be the length we need to find.

### Step 3: Set up the equation using the theorem
From the theorem, we can express the relationship as:

\[ \frac{PQ}{QS} = \frac{PR}{SR} \]

Substituting the known values:

\[ \frac{6}{3} = \frac{8}{SR} \]

### Step 4: Solve for SR
Cross-multiplying gives us:

\[ 6 \cdot SR = 3 \cdot 8 \]

This simplifies to:

\[ 6 \cdot SR = 24 \]

Dividing both sides by 6:

\[ SR = \frac{24}{6} = 4 \text{ cm} \]

### Step 5: Find QR
Now that we have SR, we can find QR using the relationship:

\[ QR = QS + SR \]

Substituting the known values:

\[ QR = 3 \text{ cm} + 4 \text{ cm} = 7 \text{ cm} \]

### Final Answer
The length of QR is **7 cm**.

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