Point D is taken in triangle ABC on side AB so that AD: DB = 2 ÷ 3. Through point D a straight line is drawn

Point D is taken in triangle ABC on side AB so that AD: DB = 2 ÷ 3. Through point D a straight line is drawn parallel to BC and intersecting AC at point K. What are the lengths of the segments AK and KC if AC = 12cm?

According to the condition of the problem, the BC is parallel to the BC. AB and AC are secants for these parallel lines. According to the generalized Thales theorem, the segments that are cut off by parallel straight lines on one secant are proportional to the segments on the other. In our case, AD / DB = 2/3, which means AK / KC = 2/3. Let the proportionality coefficient be x, then we get AK = 2x, KC = 3x.
2x + 3x = 12
5x = 12
x = 2.4
AK = 2 * 2.4 = 4.8 (cm).
KC = 3 * 2.4 = 7.2 (cm).
Answer: AK length 4.8 cm, KC length 7.2 cm.