circle
set of all points given from a starting point.
circle postulate
a circle can be drawn in a given plane with any given point as center and any given line segment as radius.
angle bisectors
the line that bisects an angle (in a triangle). all the bisectors of a triangle meet at one point. this is called being concurrent.
incenter
point equidistant from all the sides and can be used as the center of a circle, which will be inscribed inside the triangle.
median
the line segment that connects the vertex of a triangle to the midpoint of the opposite side (does not have to be an angle bisector).
centroid
2/3 of the way down the median from the vertex. center of gravity. always on the inside of the triangle.
perpendicular bisectors
the line that bisects and is perpendicular to the side of a triangle. does not have to go through the vertex.
circumcenter
equidistant from the vertices, so you can circumscribe a circle around the triangle that goes through all three vertices.right triangle: is on hypotenuse. obtuse triangle: outside the triangle.
altitude
the line that contains the vertex and is perpendicular to the opposite side or the opposite side extended.
orthocenter
intersection of altitudes. right triangle: on top of right angle.
a plane postulate corollary
in a given plane containing a line, there is one and only one perpendicular line through a given point on the line.
theorem 8
if a point is not on a line, there is only one line perpendicular to a line through the point.
perpendicular from a point to a line
if P is a point not on a line and if the line through P is perpendicular to the line and meets the line at Q then the line segment PQ is called the perpendicular from a point to a line.
projections
point Q from previous term (also known as a foot).
corollary to theorem 8
if you have a point not on a line and a line, the perpendicular line will be the shortest line from the point if you drew a lot of lines from that point.
length of a perpendicular
the distance from a point to a line is the length of the perpendicular from the point on a line.
theorem 9
a pt on the perpendicular bisector of a line segment is equidistant from the ends of the line segment.
vertical lines
a line which if extended sufficiently would pass through the center of the earth.
vertical planes
a plane which contains a vertical line.
horizontal line
a line perpendicular to a vertical line.
horizontal plane
plane determined by two intersecting horizontal lines.
infinite number of points
infinite number of points in a plane through a point.
definition of a line perpendicular to a plane
suppose a line intersects plane x at pt a. the line is perpendicular to plane x if the line is perpendicular to every member of the set of lines in the plane which contain point a.
postulate to perpendicular lines in a plane
one and only one line which contains a given point and is perpendicular to a given plane.
distance of a line
the distance from a point to a plane is the length of the perpendicular from the point to the plane.
theorem 10: Minimum Condition Theorem
minimum condition for a line to be perpendicular to a plane. if a line meets a plane at point a, and if the line is perpendicular to 2 distinct members of the set of lines in the plane which contain a, then the line is perpendicular to the plane.
def. of perpendicular planes
2 planes are perpendicular if 1 of the planes contains a line which is perpendicular to the other plane.