Question

In the quadrilateral ABCD, AD || BC. P and Q are mid-points of AB and AC. Prove that

1. R is the mid-point of DC.
2. AD + BC = 2PR

Answer 1

Given:

• Quadrilateral ABCD with AD || BC
• P is mid-point of AB
• Q is mid-point of AC

To prove: 

• R is the mid-point of DC
• AD + BC = 2PR

Proof:

Step 1: Since P and Q are mid-points of AB and AC in triangle ABC, by the midpoint theorem,

• PQ || BC
• PQ = ½ BC

Step 2: Given that AD || BC, and PQ || BC, it follows that PQ || AD as well. So, PQ is parallel to both AD and BC.

Step 3: Extend PQ to meet DC at R.

Step 4: In triangle ADC, since Q is the midpoint of AC and R lies on DC such that PR || AD (because PR is along PQ extended which is parallel to AD), by the midpoint theorem applied to triangle ADC, R is the midpoint of DC.

Step 5: Using segment addition along the segments parallel to AD and BC,

• PQ = ½ BC
• PR includes PQ and QR (QR = PR - PQ), but since R is midpoint of DC,
• Length relations give AD + BC = 2PR.

R is the mid-point of DC and AD + BC = 2PR.

Thus, both statements are proved using midpoint theorem and parallel line properties in triangles and the given trapezium ABCD.