Given: △abc, ab=bc=10 cm, ac=12 cm, ak ⊥ bc cm ⊥ ab , bn ⊥ ac . find: ak, bn, cm.

Since AB=Bc, triangle ABC is isosceles with base AC. Then BN is height and  median drawn to the base. If BN is median and AC=12 cm, then AN=NC=6 cm.

1. By the Pythagorean theorem for triangle ABN,

2. AB and BC are legs of isosceles triangle ABC. The heights drawn to the lags of isosceles triangle are congruent, so AK=CM. Consider two right triangles AKC and AKB. By the Pythagorean theorem,

Subtract from the second equation the first one:

Then

AK=CM=9.6 cm, BN=8 cm