In trapezoid ABCD, the base of BC is 12 cm. Point M does not lie in the plane of the trapezoid, and point K is the middle of the segment BM. Prove that the plane ADK intersects the MC segment at some point H, and find the segment KH.
Since ABCD is a trapezoid, BC is parallel to AD. The AC segment lies in the BMC plane, then the BMC plane is parallel to AD.
The segment AD lies in the plane ADK. Point K belongs to two planes ADK and BMC, therefore, a certain segment of KН is the intersection of the planes.
KH and BC lie in the same plane of the BMC, and since the BMC is parallel to AD, respectively, AD is parallel to BC and KH.
By condition, point K is the middle of the BM segment, then the KН segment is the middle line of the BMC triangle.
KН = BC / 2 = 12/2 = 6 cm.
Answer: The length of the KH segment is 6 cm.
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