Question

Through the mid-point Q of side CD of a parallelogram ABCD, the line AR is drawn which intersects BD at P and produced BC at R. Prove that \(AQ = QR\).

Hint:

In parallelograms, extending a side and using a midpoint often creates congruent triangles.

Answer 1

Step 1: Understanding the Concept:
To prove \(AQ = QR\), we show that \(\triangle ADQ\) and \(\triangle RCQ\) are congruent.
Step 3: Detailed Explanation:
In \(\triangle ADQ\) and \(\triangle RCQ\):
1. \(DQ = QC\) (Given that \(Q\) is the mid-point of \(CD\)).
2. \(\angle ADQ = \angle RCQ\) (Alternate interior angles, as \(AD \parallel BR\)).
3. \(\angle AQD = \angle RQC\) (Vertically opposite angles).
4. Therefore, \(\triangle ADQ \cong \triangle RCQ\) by ASA (Angle-Side-Angle) congruence rule.
5. By CPCT (Corresponding Parts of Congruent Triangles):
- \(AQ = QR\)
- \(AD = CR\)
Step 4: Final Answer:
By ASA congruence between \(\triangle ADQ\) and \(\triangle RCQ\), we find \(AQ = QR\).