PQRA is a rectangle with AP = 22 cm and PQ = 8 cm. A triangle ABC is drawn inside the rectangle so that its vertices lie on the sides of PQRA and BQ = 2 cm, QC = 16 cm along the sides meeting at Q. What is the length (in cm) of the line segment joining the midpoints of sides AB and BC of triangle ABC?

Introduction / Context:
This problem places a triangle inside a rectangle with specific distances along the sides, then asks for the length of the line segment joining the midpoints of two sides of the triangle. It tests understanding of coordinate geometry, segment midpoints, and distance calculation. By assigning coordinates to the rectangle and triangle vertices, we can find the midpoints of AB and BC and then compute the distance between these two midpoints using the distance formula.

Given Data / Assumptions:

• PQRA is a rectangle.
• AP = 22 cm and PQ = 8 cm.
• Triangle ABC is drawn with vertices on the sides of the rectangle.
• At vertex Q of the rectangle, BQ = 2 cm along one side and QC = 16 cm along the other adjacent side.
• We must find the length of the line segment joining the midpoints of AB and BC.

Concept / Approach:
We can set up a coordinate system to simplify calculations. Place the rectangle PQRA so that its vertices have convenient coordinates, then locate points B and C based on the given distances from Q along the sides. Choose A as the fourth corner of the rectangle that forms triangle ABC, so A, B, and C are three vertices of the triangle. Then we determine the midpoints of AB and BC and use the distance formula to find the length of the segment joining them. This coordinate approach avoids complicated geometry and leads to a straightforward algebraic solution.

Step-by-Step Solution:
Let us place the rectangle so that A = (0, 0), P = (8, 0), Q = (8, 22), and R = (0, 22).This makes AP = 22 cm (vertical) and PQ = 8 cm (horizontal), consistent with the given data.At vertex Q = (8, 22), side QP goes downward and side QR goes leftward.Given BQ = 2 cm along QR and QC = 16 cm along QP (or vice versa), a consistent placement is: B lies on QR and C lies on QP.Let B be 2 cm left of Q on QR: B = (8 − 2, 22) = (6, 22).Let C be 16 cm down from Q on QP: C = (8, 22 − 16) = (8, 6).Take A = (0, 0) as the rectangle corner opposite Q, so triangle ABC has vertices A(0, 0), B(6, 22), and C(8, 6).Find midpoint of AB: M = ((0 + 6) / 2, (0 + 22) / 2) = (3, 11).Find midpoint of BC: N = ((6 + 8) / 2, (22 + 6) / 2) = (7, 14).Distance MN = √[(7 − 3)^2 + (14 − 11)^2] = √(4^2 + 3^2) = √(16 + 9) = √25 = 5 cm.

Verification / Alternative check:
The coordinates of M and N are carefully derived as midpoints, and the distance calculation is straightforward. The vector from M to N is (7 − 3, 14 − 11) = (4, 3). The length of this vector is √(4^2 + 3^2) = √25 = 5, which is a familiar 3–4–5 right triangle relationship. This simple right triangle structure is another indication that the chosen coordinates and distances are correct and that the final answer is reliable.

Why Other Options Are Wrong:
4√2 cm (approximately 5.66 cm) and 3√5 cm (approximately 6.71 cm) are larger than the correct value and would arise from miscalculating the differences in coordinates or using incorrect midpoints. Values 6 cm and 10 cm are rough guesses that do not correspond to the precise distance between the midpoints given the actual coordinates. Only 5 cm exactly matches the computed distance between M and N.

Common Pitfalls:
Some students may misinterpret the placement of B and C along the sides from Q and assign incorrect coordinates. Others might confuse which corner is A when forming triangle ABC, leading to a different and incorrect triangle. Mistakes in computing midpoints or in applying the distance formula (especially squaring and adding) are also common. A clear diagram, consistent coordinate assignment, and careful arithmetic prevent these errors.

Final Answer:
The length of the segment joining the midpoints of AB and BC is 5 cm.