Note necessary facts about isosceles triangle ABC:
- The median CD drawn to the base AB is also an altitude to tha base in isosceles triangle (CD⊥AB). This gives you that triangles ACD and BCD are congruent right triangles with hypotenuses AC and BC, respectively.
- The legs AB and BC of isosceles triangle ABC are congruent, AC=BC.
- Angles at the base AB are congruent, m∠A=m∠B=30°.
1. Consider right triangle ACD. The adjacent angle to the leg AD is 30°, so the hypotenuse AC is twice the opposite leg CD to the angle A.
AC=2CD.
2. Consider right triangle BCD. The adjacent angle to the leg BD is 30°, so the hypotenuse BC is twice the opposite leg CD to the angle B.
BC=2CD.
3. Find the perimeters of triangles ACD, BCD and ABC:
[tex]P_{ACD}=AC+CD+AD=2CD+CD+AD=3CD+AD;[/tex]
[tex]P_{BCD}=BC+CD+BD=2CD+CD+AD=3CD+AD;[/tex]
[tex]P_{ABC}=AC+BC+AB=2CD+2CD+AD+BD=4CD+2AD.[/tex]
4. If sum of the perimeters of △ACD and △BCD is 20 cm more than the perimeter of △ABC, then
[tex]P_{ACD}+P_{BCD}=P_{ABC}+20,\\ \\3CD+AD+3CD+AD=4CD+2AD+20,\\ \\6CD+2AD=4CD+2AD+20,\\ \\2CD=20.[/tex]
5. Since AC=BC=2CD, then the legs AC and BC of isosceles triangles have length 20 cm.
Answer: 20 cm.
