Answer:
k = 2/5
Explanation:
Given a diagram with triangles ONM and OAB such that M is the midpoint of OB, A lies on AN with AN = 2·OA, and P is the point of intersection of NM and AB, you want the ratio AP/AB.
Proportion
Please refer to the attached diagram.
Point C is located at the midpoint of AN, which makes OA≅AC≅CN, or AN is 2/3 of ON. Segments AD and CE are parallel to AN, so divide OM into thirds. The length DM is 2 of those thirds, and the length MB is equal to OM, so is 3 of those thirds. That is, the ratio DM/DB is (2/3)/(2/3+3/3) = 2/5.
Triangle BAD is similar to triangle BPM, so the ratio AP/AB is also 2/5.