In isosceles triangle PQR, sides PQ and PR are each 18 cm. Point A is the midpoint of base QR. Through A, segment AB is drawn parallel to PR and meets PQ at B, and segment AC is drawn parallel to PQ and meets PR at C. What is the perimeter of quadrilateral ABPC in centimetres?

Introduction / Context:
This problem tests understanding of midpoints, parallel lines and similar triangles inside an isosceles triangle. By drawing lines through the midpoint of the base parallel to the equal sides, we create a smaller quadrilateral inside the triangle. The question asks for the perimeter of this quadrilateral. Recognising that several segments are mid segments and that the quadrilateral turns out to be a rhombus or parallelogram allows us to determine all its side lengths using simple proportional reasoning.

Given Data / Assumptions:
- Triangle PQR is isosceles with PQ = PR = 18 cm. - A is the midpoint of QR, so QA = AR. - Line AB is drawn through A parallel to PR, meeting PQ at B. - Line AC is drawn through A parallel to PQ, meeting PR at C. - We must find the perimeter of quadrilateral ABPC.

Concept / Approach:
Because AB is parallel to PR and A lies on QR, point B becomes the midpoint of PQ by the midpoint theorem: a line segment through the midpoint of one side of a triangle and parallel to another side meets the third side at its midpoint. Similarly, since AC is parallel to PQ and A lies on QR, point C becomes the midpoint of PR. This means A, B and C are midpoints of QR, PQ and PR respectively. Segments AB and AC are mid segments of triangle PQR, so each has length equal to half of the corresponding side. The quadrilateral ABPC has four equal sides and opposite sides parallel, so it is a rhombus with side length equal to half the original equal sides.

Step-by-Step Solution:
Step 1: Since A is midpoint of QR and AB ∥ PR, triangle ABQ is similar to triangle PRQ. Step 2: By the midpoint theorem, B is the midpoint of PQ and AB is half of PR. Step 3: Similarly, AC ∥ PQ with A midpoint of QR implies C is midpoint of PR and AC is half of PQ. Step 4: Given PQ = PR = 18 cm, their halves are PB = 9 cm and PC = 9 cm. Step 5: As mid segments, AB = PR / 2 = 9 cm and AC = PQ / 2 = 9 cm. Step 6: Quadrilateral ABPC has sides AB, BP, PC and CA, each equal to 9 cm. Step 7: Therefore the perimeter of ABPC is 4 * 9 = 36 cm.

Verification / Alternative check:
We can also place triangle PQR in a coordinate system with Q and R on the horizontal axis and P above the base. Choosing Q at (−b, 0), R at (b, 0) and P at (0, h), point A is at (0, 0), the midpoint of QR. Lines through A parallel to the equal sides intersect PQ and PR at points that turn out to be midpoints B and C. Computing distances confirms AB = BP = PC = CA and each equals half of the original equal side. This geometric and coordinate reasoning both agree that the quadrilateral is a rhombus of side 9 cm, giving perimeter 36 cm.

Why Other Options Are Wrong:
Option 18 would correspond to the sum of only two sides of the rhombus, not all four. Option 28 does not relate to any simple fraction of 18 and ignores the symmetry of the construction. Option 32 again does not correspond to a multiple of half the original side length and has no geometric basis in this setup.

Common Pitfalls:
Some learners do not immediately see that B and C are midpoints and may attempt to assign arbitrary coordinates and perform heavy algebra. Others might assume ABPC is a rectangle and misjudge side lengths. Forgetting the midpoint theorem or misunderstanding which segments are parallel can also lead to incorrect reasoning. Drawing a clear diagram and marking midpoints and parallel lines makes the relationships much easier to see.

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