In `DeltaABC, /_A: /_B: /_C = 3:3:4`. A line parallel to BC is drawn which touches AB and AC at P and Q respectively. What is the value of `/_AQP- /_APQ`?

Options

A. 12
B. 18
C. 24
D. 36

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the Angle Ratios
Given that the angles in triangle ABC are in the ratio \(3:3:4\), we can denote the angles as:
- \( \angle A = 3k \)
- \( \angle B = 3k \)
- \( \angle C = 4k \)

### Step 2: Set Up the Equation
The sum of the angles in a triangle is \(180^\circ\). Therefore, we can write the equation:
\[ 3k + 3k + 4k = 180^\circ \]
This simplifies to:
\[ 10k = 180^\circ \]

### Step 3: Solve for \(k\)
To find the value of \(k\), divide both sides of the equation by 10:
\[ k = \frac{180^\circ}{10} = 18^\circ \]

### Step 4: Calculate Each Angle
Now we can calculate the angles:
- \( \angle A = 3k = 3 \times 18^\circ = 54^\circ \)
- \( \angle B = 3k = 3 \times 18^\circ = 54^\circ \)
- \( \angle C = 4k = 4 \times 18^\circ = 72^\circ \)

### Step 5: Identify Corresponding Angles
Since a line parallel to BC is drawn, angles \( \angle APQ \) and \( \angle AQP \) correspond to angles \( \angle B \) and \( \angle C \) respectively:
- \( \angle APQ = \angle B = 54^\circ \)
- \( \angle AQP = \angle C = 72^\circ \)

### Step 6: Calculate the Difference
Finally, we need to find the value of \( \angle AQP - \angle APQ \):
\[ \angle AQP - \angle APQ = 72^\circ - 54^\circ = 18^\circ \]

### Final Answer
Thus, the value of \( \angle AQP - \angle APQ \) is \( \boxed{18^\circ} \).

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