In triangle PQR, PQ = PR = 18 cm. Point A is the mid-point of QR. Segments AB and AC are drawn parallel to PR and PQ respectively. What is the perimeter, in centimetres, of quadrilateral ABPC?

Introduction / Context:
This geometry question involves an isosceles triangle and a quadrilateral formed by drawing lines parallel to the sides. It tests understanding of similar triangles and symmetry. The triangle PQR has two equal sides, and a mid point on the base is used together with parallels to create a smaller quadrilateral ABPC inside. The task is to find the perimeter of this quadrilateral.

Given Data / Assumptions:
• Triangle PQR is isosceles with PQ = PR = 18 cm.• A is the mid point of QR.• AB is drawn through A parallel to PR.• AC is drawn through A parallel to PQ.• Points B and C lie on PQ and PR respectively.• Quadrilateral ABPC is formed, and we need its perimeter.

Concept / Approach:
Because PQR is isosceles, the configuration is symmetric about the altitude from P to QR. Point A is at the centre of the base, and lines AB and AC are drawn parallel to the equal sides. Using coordinate geometry or symmetry, we find that AB, BC, CP, and PA are all equal in length. In particular, each side of quadrilateral ABPC turns out to be 9 cm, which is half of the equal sides in the original triangle. Therefore, the perimeter is four times this common side length.

Step-by-Step Solution:
Step 1: Represent triangle PQR in a coordinate system for clarity.Let QR be horizontal with Q at (−a, 0) and R at (a, 0). Let P be at (0, h). The equal sides PQ and PR each have length 18 cm.Step 2: Since A is the mid point of QR, A is at (0, 0).Step 3: The line through A parallel to PR will intersect PQ at B, and the line through A parallel to PQ will intersect PR at C.Step 4: Using similarity of triangles or direct coordinate calculations, one can show that the segments AB, BP, PC, and CA are all equal and each equals 9 cm.Step 5: Hence quadrilateral ABPC has four equal sides each of 9 cm.Step 6: Perimeter of ABPC = 4 * 9 = 36 cm.

Verification / Alternative check:
A more algebraic approach confirms the same result. Taking concrete values for the base and height that produce side length 18 cm and solving using the section formula and parallel line properties, we again find each side of the quadrilateral is 9 cm. In either method, the symmetry of the isosceles triangle and the use of parallels from the base midpoint ensure the quadrilateral behaves like a smaller rhombus inside the triangle.

Why Other Options Are Wrong:
Option A: 18 would correspond to all four sides being 4.5 cm each, which does not match the geometry.Option B: 28 gives an average side length of 7 cm, inconsistent with the similarity ratio derived from the original triangle.Option C: 32 implies each side is 8 cm, which is still not half of 18 cm and does not arise from the correct construction.

Common Pitfalls:
Students may attempt to compute side lengths directly from incomplete geometric reasoning or may incorrectly assume that some sides of the quadrilateral match either the base or the original sides. Others might overlook the symmetry and not recognise that all four sides are equal. Drawing a clear labelled diagram and using the properties of parallel lines and mid points in an isosceles triangle helps reach the correct conclusion.

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