Point M is the middle of the lateral side of the CD trapezoid ABCD. Line BM intersects line AD

Point M is the middle of the lateral side of the CD trapezoid ABCD. Line BM intersects line AD at point K. Find the length of the line segment AK if the bases of the trapezoid are 5 and 12 cm.

Let us prove that the triangles BCM and DCM are similar.

Angle BMC = DMC as vertical angles at the intersection of straight lines BK and CD.

Angle СBМ = DCM as criss-crossing angles at the intersection of parallel straight lines AK and BC of the secant BK.

Then the triangles BCM and DKM are similar in two angles.

Since, according to the condition, point M is the middle of CD, then CM = DM, and the coefficient of similarity of triangles BCM and DKM is 1.

Then DK = BC = 5 cm.

Segment AK = AD + DK = 12 + 5 = 17 cm.

Answer: The length of the segment AK = 17 cm.