The segments CD and AB intersect at point O so that CO = DO, AC is parallel to BD. The perimeter of the triangle

The segments CD and AB intersect at point O so that CO = DO, AC is parallel to BD. The perimeter of the triangle BOD = 22 cm, CD = 18 cm, the segment AO is 3 cm shorter than BD. Find the length of the segment AC.

Consider triangles AOC and ODB. They have:

СО = ОD by condition, angle СОА = angle DOB as vertical, angle АСО = angle ODB as criss-crossing for AC II BD and secant CD. Consequently, the triangles are equal in attribute II.

Hence, AO = OB.

Let BD = x, then AO = x – 3 (by hypothesis). And since AO = ОВ, hence ОВ = х – 3

x + x – 3 + 9 = 22

2x = 16

x = 8cm.

AC = BD = 8 cm (from the equality of triangles)

Answer: segment AC = 8 cm.