Question

In triangle ABCD, where AB=BC and DC is parallel to AB. Perpendiculars DE and CF are drawn. If triangle BCF is congruent to triangle BCD, what is the relationship between DE and CF?

a) DE = CF
b) DE < CF
c) DE > CF
d) Insufficient information

Answer 1

In the given quadrilateral, where AB is parallel to DC, and triangle BCF is congruent to triangle BCD, the perpendiculars DE and CF are equal in length. Therefore, DE equals CF.

Step-by-step explanation:

The question relates to determining the relationship between the lengths of perpendiculars DE and CF from points D and C of a quadrilateral to the base AB, given that triangle BCF is congruent to triangle BCD, and that AB is parallel to DC.

Since AB=BC and triangles BCF and BCD are congruent, it follows that triangle BCF is an isosceles right triangle, meaning that the lengths of the legs are equal (BC=CF). Considering that DC is parallel to AB, angle DBC is congruent to angle FCB, both being right angles. As angle CBD is congruent to itself by the reflexive property of congruence, triangles BCD and BCF are congruent by the Angle-Side-Angle postulate. Since DE is perpendicular to AB and forms a right angle, triangle BCD is also a right angle triangle with BD as the hypotenuse.

Due to the congruence of triangles BCF and BCD, the sides CD and CF are equal, and since DE is an altitude of BCD, DE and CF must be equal as well, which makes the correct answer (a) DE = CF.