In or Out Investigation

By

Scottie Benford

Each triangle has three altitudes. The diagram above illustrates three red altitudes of a particular triangle. Given any triangle, do the altitudes lie on a side, on the inside, or on the outside of the triangle? How do your results depend on different types of triangles.

To be able to illustrate my points I will use three types of triangles acute, right, and obtuse.   My first construction will be of an acute triangle.  All of my constructions will be done using GSP.

This is my first construction.  This is a construction of the altitudes of an acute triangle.  I constructed my triangle by using three points and connecting the points with segments.  My points were labeled A, B, and C.  Next I constructed an altitude from vertex B to segment AC by using a perpendicular line.  Next I constructed an altitude from vertex A to segment BC using a perpendicular line.  I followed the same format and constructed an altitude from vertex C to segment AB.  From my construction I concluded that the altitudes of an acute triangle are all inside.

My next construction is of a right angle.  The angle was constructed by using two points to make a segment.  After constructing the segment I used the line tool to construct a segment that would meet the first segment perpendicularly.  I highlighted my two points and then constructed the last segment to complete my right triangle. My points were labeled as A, B, and C.  I used the measuring tool to make sure that I had at least one right angle inside my triangle.   My first altitude was constructed from vertex B to segment AC by using a perpendicular line.   Altitude two was constructed from vertex C to segment BA.  The altitude is exact same as the base of the triangle.  The third altitude was constructed by using a perpendicular line from vertex A to segment BC.  This altitude was also part of the side of the triangle.  From this construction I have concluded that when using a right triangle only one altitude will be inside the triangle.  The other two altitudes, one being the height and the other being the base, are part of or go along the segments that make up the triangle.

My final construction is that of an obtuse triangle.  I constructed this triangle by using the segment tool.  After constructing the triangle I labeled the points A, B, and C.  I constructed my first altitude from vertex B to segment AC by using a perpendicular line.  My second altitude was constructed by using a perpendicular line from vertex A to segment BC.  This altitude was totally outside the triangle but I do understand that if segment BC was extended it would cross.  The third altitude was constructed by using a perpendicular from vertex C to segment BA.  This is a similar situation to the altitude from vertex A.  The altitude is totally outside of the circle and would cross segment BA if it were extended.  From my construction I have concluded that when observing the altitudes of an obtuse triangle only one will be inside and two will be outside.  Overall I feel that the wider an angle of a triangle becomes the more movement an altitude will have towards the outside of the triangle.