Mathematics

Mathematics

Problem

In △ABC\triangle ABC, AB=ACAB = AC. Point DD lies on BCBC such that AD⊥BCAD \perp BC. Answer the following questions. a. (i) Prove that △ABD≅△ACD\triangle ABD \cong \triangle ACD. b. (ii) Using the result of part (i), show that BD=DCBD = DC. c. (iii) If BC=12 cmBC = 12\,\text{cm}, find the lengths BDBD and DCDC.
Intuitive explanation

Since △ABC\triangle ABC is isosceles with AB=ACAB = AC, the altitude from AA to the base BCBC has a special property: it's also the perpendicular bisector of BCBC. This means DD isn't just the foot of the altitude—it's the midpoint of BCBC, and the altitude is a line of symmetry for the entire triangle. Use this symmetry to unlock relationships between the segments and angles.

Intuitive explanation

Since △ABC\triangle ABC is isosceles with AB=ACAB = AC, the altitude from AA to the base BCBC has a special property: it's also the perpendicular bisector of BCBC. This means DD isn't just the foot of the altitude—it's the midpoint of BCBC, and the altitude is a line of symmetry for the entire triangle. Use this symmetry to unlock relationships between the segments and angles.

Browse the problem library