Geometry Chapter 5

Midsegment of a triangle

A segment formed between the midpoints of two sides of a triangle

Theorem 5.1

Tells us that the mid segment is parallel to the third side and half the length of that third side

Perpendicular bisector

A line/race/segment formed through the midpoint of a segment and at a right triangle to that segment

Theorem 5.2

If a point is on the perpendicular bisector, then it is equidistant from the end points of the bisected segment

Theorem 5.3 (converse of 5.2)

If a point is equidistant from the end point of bisected segment a point is on a perpendicular bisector then it is a point on a perpendicular bisector

Concurrency

If lines all contain a single point

Theorem 5.4

The perpendicular bisectors of a triangle are concurrent at a point called the circumcenter

Circumcenter

Where are the perpendicular bisectors are concurrent

Angle bisector

A line/Ray/segment formed through the vertex of an angle, which cuts the angle in equal halves

Theorem 5.5

If a point is on the angle, bisector that is equidistant from the two sides of the angle

Theorem 5.6 (converse to theorem 5.5)

If a point is equidistant from the two sides of the angle, then it is a point on the angle bisector

Theorem 5.7

The angle bisector of a triangle are concurrent at a point called the incenter the incenter is the center of the inscribed circle

incenter

The point where the angle bisectors meet

Median

A segment formed from a vertex of a triangle to the midpoint of the opposite

Theorem 5.8 concurrency of medians

The medians are concurrent at a point called the centroid, which is 2/3 of the median length from the vertex

Altitude

A perpendicular segment formed from a vertex of a triangle to opposite side

Theorem 5.9 concurrency of altitudes

The altitudes are concurrent at a point called the ortho center

Theorem 5.10

If one side of a triangle is longer than another side, and the angle opposite the longer side is a larger angle

Theorem 5.11

If one angle of a triangle is larger than another angle than the side tended by the larger angle is the larger side

Theorem 5.12 triangle inequality theorem

The sum of two side sides of a triangle must be greater than the length of the third side

Theorem 5.13 hinge theorem

If two sides of one triangle concurrent did two sides of one triangle, and they included side of the first is larger than they included out of the second and the third side of the first triangles longer than the third side of the Second

Theorem 5.14 converse to hinge theorem

Two sides of one congruent to two sides of one triangle third side of the triangles longer than the third side of the second included angle of first is larger than the second