Mathematics

Mathematics

Problem

In an isosceles triangle ABCABC, AC=BCAC = BC. Point DD is the midpoint of ABAB. (i) List the three pairs of equal sides between △ACD\triangle ACD and △BCD\triangle BCD. (ii) Prove that △ACD≅△BCD\triangle ACD \cong \triangle BCD. (iii) Deduce that CDCD is perpendicular to ABAB.
Intuitive explanation

The key insight is that the line from the apex of an isosceles triangle to the midpoint of the base creates a natural mirror symmetry. Since AC=BCAC = BC and DD bisects ABAB, the two smaller triangles share the segment CDCD and have matching side lengths by construction—this symmetry forces them to be congruent, which then reveals that CDCD must be perpendicular to the base.

Intuitive explanation

The key insight is that the line from the apex of an isosceles triangle to the midpoint of the base creates a natural mirror symmetry. Since AC=BCAC = BC and DD bisects ABAB, the two smaller triangles share the segment CDCD and have matching side lengths by construction—this symmetry forces them to be congruent, which then reveals that CDCD must be perpendicular to the base.

Browse the problem library