In `Delta ABC, PQ` is parallel to `BC`. If `AP : PB = 1 : 2 `and `AQ = 3 cm`, AC is equal to

Options

A. 6 cm
B. 9 cm
C. 12 cm
D. 8 cm

Answer

To solve the problem step by step, we will use the Basic Proportionality Theorem (also known as Thales' theorem).

### Step 1: Understand the given information
We have triangle ABC with a line PQ parallel to side BC. The ratio of segments AP to PB is given as 1:2, and the length of segment AQ is given as 3 cm.

### Step 2: Set up the ratio
According to the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Therefore, we can write the following ratio based on the given information:

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

Given that \( AP : PB = 1 : 2 \), we can express this as:

\[ \frac{1}{2} = \frac{AQ}{QC} \]

### Step 3: Substitute the known value
We know that \( AQ = 3 \, \text{cm} \). We can substitute this value into the ratio:

\[ \frac{1}{2} = \frac{3}{QC} \]

### Step 4: Solve for QC
To find QC, we can cross-multiply:

\[ 1 \cdot QC = 2 \cdot 3 \]
\[ QC = 6 \, \text{cm} \]

### Step 5: Find the length of AC
Now that we have both AQ and QC, we can find the total length of AC:

\[ AC = AQ + QC = 3 \, \text{cm} + 6 \, \text{cm} = 9 \, \text{cm} \]

### Final Answer
Thus, the length of AC is \( 9 \, \text{cm} \).

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