Question

Let ABCD be a convex quadrilateral such that angle DAB is acute. Angle ADB and Angle ACB are complementary angles. Angle DBC and 2*Angle DBA are supplementary angles. Show that (DB+BC)^2=AD^2+AC^2

Answer 1

Answe

extend CB such that DB=BE

since 2x+y=180(given)

so angle ABC=x

triangle EBA is congruent to triangle DBA(SAS)

so AE=AD

angleE=angleD

angleA=angle CAE=90(angle sum property)

CE^2=AC^2+AE^2

(CB+BE)^2=AC^2+AE^2

(BC+BD)^2=AC^2+AD^2