NAME:_______________________________
Geometry: Concurrence of Altitudes, Medians, Angle Bisectors, and Perpendicular Bisectors
Mrs. Gilbert
Old definitions:
1. Median: a segment that starts at a vertex of a triangle and crosses the opposite side at its midpoint
2. Altitude: a segment that starts at a vertex of a triangle and is perpendicular to the opposite side
3. Angle Bisector: a segment that starts at a vertex and divides the vertex angle into 2 equal angles
4. Perpendicular Bisector: a line that perpendicular to a line segment and intersects at the segment’s
midpoint. (they do not necessarily go through a vertex of a triangle)
New definitions:
1. Concurrent: at the same time (concurrent lines intersect at the same point)
2. Centroid:a point where the 3 medians of a triangle intersect - It divdes the medians into a ratio of 2:1
Point P is the Centroid of triangle AEC.
3. Orthocenter: a point where the 3 altitudes of a triangle intersect.
4.
Incenter: a point where the 3 angle bisectors of a triangle intersect.
Point D is the Incenter of triangle ABC
D
5.
Circumcenter: a point where the 3 perpendicular bisectors of a triangle’s sides intersect.
Point P is the Circumcenter of triangle ABC
** In EQUILATERAL triangles, the concurrency points are the same point.
** In ISOSCELES triangles, the concurrency points are collinear.
** A special property of the Centroid:
The Centroid of a triangle divides the medians into a ratio of 2:1 where the part of the
median that is double the other segment is always the piece that goes from the vertex of the triangle to
the centroid.
Problems to Solve:
1.
̅̅̅̅ 𝒂𝒓𝒆 𝒎𝒆𝒅𝒊𝒂𝒏𝒔 𝒊𝒏 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆 𝑸𝑹𝑺.
2. ̅̅̅̅
𝑩𝑹, ̅̅̅̅
𝑸𝑨, 𝒂𝒏𝒅 𝑺𝑪
a.
If RP = 11, find RB.
b. If QP = 9, find PA.
c. If SP = 2x and SC = 5x - 4, find SP and SC.
d. If PB = 3x and PR = 7x-5, find PB and PR.
e. If PA=2x+6 and QA = 10x, find PA and QA.
The coordinates of the vertices of triangle ABC are A(-2,-2), B(0,4), and C(3,-2).
3. Find the coordinates of the centroid of triangle ABC.
4. Find the coordinates of the orthocenter of triangle ABC.
5. Find the coordinates of the circumcenter of triangle ABC.
6. Can you tell what type of triangle, triangle ABC is? If so explain or show work.