Answer:
20 cm.
Step-by-step explanation:
The perimeter of triangle
- ACD is [tex]P_{ACD}=AC+CD+AD;[/tex]
- BCD is [tex]P_{BCD}=BC+CD+BD;[/tex]
- ABC is [tex]P_{ABC}=AB+BC+AC=AD+DB+AC+BC.[/tex]
Since the sum of the perimeters of △ACD and △BCD is 20 cm more than the perimeter of △ABC, you have that
[tex]AC+CD+AD+BC+CD+BD=AD+BD+AC+BC+20,\\ \\2CD=20,\\ \\CD=10\ cm.[/tex]
Consider right triangle ACD. In this triangle [tex]\angle A=30^{\circ},[/tex] then the hypotenuse AC is twice the leg CD. Hence, [tex]AC=2\cdot 10=20\ cm.[/tex]
