In an acute-angled triangle MNP, the bisector of angle M intersects the height NK at point O.

| September 7, 2021 | education

In an acute-angled triangle MNP, the bisector of angle M intersects the height NK at point O. Moreover, OK = 9cm. Find the distance from point O to line MN.

MS is the bisector of the angle M.
The distance from a point to a straight line is the perpendicular dropped from a point to a straight line. Let the height OH be the distance from point O to straight line MN.
Consider triangles MHO and MCO.
In the MTH triangle: angle ОМH = angle М / 2 (since MC is a bisector), angle МHО = 90 degrees (since OH is height).
Consider the MCO triangle: angle OMC = angle M / 2 (since MC is a bisector), angle MCO = 90 degrees (since OH is height).
The MO side is the common side of both triangles.
Consequently, the triangles MHO and MKO are equal in side and two adjacent angles.
Therefore, OH = OK = 9 cm.
Answer: OH = 9 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.