Geometry Chapter 5 (Properties and Attributes of a Triangle)

http://slowellgeom.blogspot.com/ -  1st part of sammi lowells blog is on this link

5.1
Vocabulary
1. Equidistant - When the point is the same distance from two or more objects
2. Locus - A set of points the satisfies a given condition

Therorems

1. Perpendicular Bisector Theorem - if a point is on the perpendicular bisector of a segment, then it is equidistant form the endpoints of the segment
2. Converse of the Perpendicular Bisector Theorem - if a point is equidistant from th endpoints of a segment,then it is on the perpendicular bisector of the segment
3. Angle Bisector Theorem - If a point is on the bisector of an angle, then it is eqidistant from the sides of the angle





5.2

Vocabulary

Concurrent - when 3 or more lines intersect at one point
Point of concurrecy  - The point where they all intersect
Circumcenter of the triangle - The point of concurrency
Circumscribed  - a circle that contains all verticies of a polygon
Incenter of the triangle - Point of concurrency
Inscribed - in a polygon intersects each line that caontains a side of the polygon at exactly one point


Theorems
1. Circumcenter theorem - the circumcenter of a trinagle is equidistant from the verticies of a triangle
2. Incenter theorem - the incenter of a trangle is equidistant from the sides of a the triangle




5.3

Vocabulary

Median of a triangle - a segment whose endpoints are a vertex of the triangle and midpoint of the other side
Centroid of a triangle - point of concurrency and of the medians of a triangle
Altitude of a triangle - A perpendicular segment from a vertex to the line containing the opisite side
Orthocenter of a triangle - Point of concurrency

Theorems

Centroid theorem - The centroid  is located 2/3 of the distance from each vertex to the midpoint of the opposite side



5.4

Vocabulary
midsegment of a triangle - a segment that joins the midpoint of two sides of a triangle


Theorems

1. triangle midsegment theorem - A midsegment of a triangle is parellel to a side of the triangle, and its length

5.5

Vocabulary
indirect proof - begin assuming that the conclusion is false the show that the assumption leads to contradiction

Theorems

Angle-side relationship theorem - if two sides of a triangle are not congruent , the the larger angle is opposite to the longer side

Trianlge inequality theorem - the sum of any two lengths of a triangle is greater then the third side length
AB+BC=AC

5.6

Theorems

Hinge Theorem - if two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle

m<A > m <D        BC>EF

5.7

Vocabulary

Pythagorean triple -    A2 + B2 = C2

5.8

Theorems

45, 45, 90 triangle theorem - Both legs are congruent and the length of the hypotenuse is the length of a leg squared rooted

30, 60, 90 triangle theorem - the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg time 3 square rooted