Problem:
In convex quadrilateral KLMN side MN is perpendicular to diagonal KM, side KL is perpendicular to diagonal LN,MN=65, and KL=28. The line through L perpendicular to side KN intersects diagonal KM at O with KO=8. Find MO.
Solution:
Let P be the intersection of line LO with line KN. Then △KPL∼△KLN, so KP28​=28KN​ and KP⋅KN=282. Also △KPO∼△KMN, so 8KP​= KN8+MO​ and KP⋅KN=8(8+MO). Thus 282=8(8+MO), from which MO=90.
Note that the value of MN is irrelevant.
The problems on this page are the property of the MAA's American Mathematics Competitions
