Question

question 7 points 4
show that the given quadrilateral pqrs is a parallelogram. give the reason.
(image of quadrilateral pqrs with diagonals intersecting at a, segments marked: qa=6, ra=9, pa=9, sa=6)

Explanation:

Step1: Identify Diagonal Segments

In quadrilateral \(PQRS\), diagonals \(PR\) and \(QS\) intersect at \(A\). From the diagram, \(PA = 9\), \(AR = 9\), so \(PA = AR\) (diagonal \(PR\) is bisected). Also, \(QA = 6\), \(AS = 6\), so \(QA = AS\) (diagonal \(QS\) is bisected).

Step2: Apply Parallelogram Theorem

A quadrilateral is a parallelogram if its diagonals bisect each other. Since \(PA = AR\) and \(QA = AS\), diagonals \(PR\) and \(QS\) bisect each other at \(A\). By the theorem, \(PQRS\) is a parallelogram.

Answer:

Quadrilateral \(PQRS\) is a parallelogram because its diagonals \(PR\) and \(QS\) bisect each other ( \(PA = AR\) and \(QA = AS\) ), and a quadrilateral with bisecting diagonals is a parallelogram.