Point D is taken in triangle ABC on side a AB so that AD: DB = 2: 3. A straight line is drawn through point D, parallel to BC and intersecting AC at point K. What are the lengths of the segments AK and KC if AC = 12?
Since the segment DK is parallel to the segment BC, then, according to Thales’ theorem, the segments cut off on the AB side are equal to the segments cut off on the AC side.
Then, since AD / BD = AK / CK = 2/3.
Let the length of the AK = 2 * X cm, then the length of the segment CK = 3 * X cm.
Then 2 * X + 3 * X = 12.
5 * X = 12.
X = 12/5 = 2.4 cm.
AK = 2 * 2.4 = 4.8 cm.
CK = 3 * 2.4 = 7.2 cm.
Answer: The length of the segment AK = 4.8 cm, CK = 7.2 cm.
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