Question

2 in quadrilateral badc, ab = ad and bc = dc. the line ac is a line of symmetry for this quadrilateral. a. based on the line of symmetry, explain why the diagonals ac and bd are perpendicular.

Explanation:

Since \(AC\) is a line of symmetry, reflecting the quadrilateral over \(AC\) maps \(B\) to \(D\). Let the intersection of \(AC\) and \(BD\) be \(O\). Then \(BO = OD\) (because of symmetry). Also, \(\triangle ABO\) and \(\triangle ADO\) are congruent (by SSS: \(AB = AD\), \(BO=OD\), \(AO = AO\)). So \(\angle AOB=\angle AOD\). Since \(\angle AOB+\angle AOD = 180^{\circ}\) (they form a linear pair), then \(\angle AOB=\angle AOD = 90^{\circ}\). Thus \(AC\perp BD\).

Answer:

The diagonals \(AC\) and \(BD\) are perpendicular because when reflecting over the line of symmetry \(AC\), \(B\) maps to \(D\). The intersection point \(O\) of \(AC\) and \(BD\) has \(BO = OD\). Congruent triangles \(\triangle ABO\) and \(\triangle ADO\) (by SSS) give \(\angle AOB=\angle AOD\). Since \(\angle AOB+\angle AOD = 180^{\circ}\), each is \(90^{\circ}\), so \(AC\perp BD\).