In `DeltaPQR, PQ=PR=18`cm, AB and AC are parallel to lines PR and PQ respectively. If A is the mid-point of QR, then what is the perimeter (in cm) of quadrilateral ABPC?

Options

A. 18
B. 28
C. 32
D. 36

Answer

To solve the problem, we need to find the perimeter of quadrilateral ABPC in triangle PQR where PQ = PR = 18 cm and A is the midpoint of QR. Lines AB and AC are parallel to PR and PQ, respectively.

### Step-by-Step Solution:

1. **Draw Triangle PQR**:
- Start by drawing triangle PQR with points P, Q, and R.
- Since PQ = PR = 18 cm, label PQ and PR as 18 cm each.

2. **Identify Midpoint A**:
- Mark point A as the midpoint of side QR. This means that the length of QA = AR.

3. **Draw Parallel Lines**:
- Draw line AB parallel to line PR and line AC parallel to line PQ. Since A is the midpoint of QR, lines AB and AC will also divide the triangle into smaller triangles.

4. **Apply Midpoint Theorem**:
- By the midpoint theorem, since A is the midpoint of QR, the length of AB will be half of PR.
- Therefore, \( AB = \frac{1}{2} \times PR = \frac{1}{2} \times 18 \text{ cm} = 9 \text{ cm} \).

5. **Calculate Length of AC**:
- Similarly, since A is the midpoint of QR and AC is parallel to PQ, the length of AC will also be half of PQ.
- Thus, \( AC = \frac{1}{2} \times PQ = \frac{1}{2} \times 18 \text{ cm} = 9 \text{ cm} \).

6. **Determine Lengths of PB and PC**:
- Since A is the midpoint of QR, and AB and AC are parallel to the sides of the triangle, the lengths PB and PC will be equal to the lengths of AB and AC respectively.
- Therefore, \( PB = 9 \text{ cm} \) and \( PC = 9 \text{ cm} \).

7. **Calculate the Perimeter of Quadrilateral ABPC**:
- The perimeter of quadrilateral ABPC can be calculated by adding the lengths of all its sides:
\[ \text{Perimeter} = AB + BP + PC + AC = 9 \text{ cm} + 9 \text{ cm} + 9 \text{ cm} + 9 \text{ cm} = 36 \text{ cm}. \]

### Final Answer:
The perimeter of quadrilateral ABPC is **36 cm**.