Inside a triangle ABC, a straight line parallel to BC intersects AB and AC at the point P and respectively. If `AB = 3 PB`, then `PQ : BC` is

Options

A. a) `1:3`
B. b) `3:4`
C. c) `1:2`
D. d) `2:3`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript:

### Step 1: Draw Triangle ABC
We start by drawing triangle ABC, where we label the vertices A, B, and C.

### Step 2: Draw Line PQ Parallel to BC
Next, we draw a line PQ that is parallel to side BC of triangle ABC. This line intersects AB at point P and AC at point Q.

### Step 3: Identify Given Information
We are given that \( AB = 3 \cdot PB \). This means that the length of segment AB is three times the length of segment PB.

### Step 4: Establish Ratios Using Similar Triangles
Since line PQ is parallel to side BC, triangles APQ and ABC are similar by the Angle-Angle (AA) similarity criterion. This gives us the following ratio:
\[ \frac{AP}{AB} = \frac{PQ}{BC} \]

### Step 5: Express AP in Terms of PB
From the given information, we can express \( AB \) in terms of \( PB \):
\[ AB = AP + PB \]
Given \( AB = 3 \cdot PB \), we can substitute:
\[ 3 \cdot PB = AP + PB \]
Rearranging gives:
\[ AP = 3 \cdot PB - PB = 2 \cdot PB \]

### Step 6: Substitute AP into the Ratio
Now we substitute \( AP \) back into the ratio we established:
\[ \frac{AP}{AB} = \frac{2 \cdot PB}{3 \cdot PB} \]
This simplifies to:
\[ \frac{AP}{AB} = \frac{2}{3} \]

### Step 7: Relate PQ and BC
Now, using the similarity of triangles:
\[ \frac{PQ}{BC} = \frac{2}{3} \]

### Step 8: Final Ratio
Thus, we can conclude that:
\[ PQ : BC = 2 : 3 \]

### Final Answer
The ratio \( PQ : BC \) is \( 2 : 3 \).

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