If P and Q are points on the sides AB and AC respeactively of `triangleABC`, if PQ||BC,l AP= 2 cm , AB = 6 cm and AC= 9 cm find AQ.

Answer

To solve the problem step by step, we will use the properties of similar triangles.

### Step 1: Understand the given information
We have triangle ABC with points P and Q on sides AB and AC respectively. We know:
- \( AP = 2 \, \text{cm} \)
- \( AB = 6 \, \text{cm} \)
- \( AC = 9 \, \text{cm} \)
- \( PQ \parallel BC \)

### Step 2: Use the properties of similar triangles
Since \( PQ \parallel BC \), triangles \( APQ \) and \( ABC \) are similar by the Basic Proportionality Theorem (also known as Thales' theorem). This gives us the following ratio:

\[ \frac{AP}{PB} = \frac{AQ}{QC} \]

### Step 3: Find PB
We can find \( PB \) using the relationship:
\[ PB = AB - AP = 6 \, \text{cm} - 2 \, \text{cm} = 4 \, \text{cm} \]

### Step 4: Set up the ratio
Let \( AQ = y \) and \( QC = AC - AQ = 9 \, \text{cm} - y \). Now we can write the ratio:

\[ \frac{AP}{PB} = \frac{AQ}{QC} \implies \frac{2}{4} = \frac{y}{9 - y} \]

### Step 5: Cross-multiply to solve for y
Cross-multiplying gives us:

\[ 2(9 - y) = 4y \]

Expanding this:

\[ 18 - 2y = 4y \]

### Step 6: Combine like terms
Now, combine the terms involving \( y \):

\[ 18 = 4y + 2y \implies 18 = 6y \]

### Step 7: Solve for y
Dividing both sides by 6 gives:

\[ y = \frac{18}{6} = 3 \, \text{cm} \]

### Step 8: Conclusion
Thus, \( AQ = 3 \, \text{cm} \).