geometry exam

isometry

maps from the plane to itself that preserves distances between points

collection of isometrys that fix a equilateral triangle to itself (triangle ABC to A’B’C’ where those are the midpoints)

the identity, reflections about about AA’, BB’, CC’, rotations around center D by 120 and 240

3 main types of isometry

reflection, rotations, translations, (the identity)

translation

direct and either f is the identity or f fixes NO points → DIRECT and fixes NO POINTS

reflection

through a line L if f(P)=P on L but F(Q) does not equal Q on line L →NOT DIRECT and fixes ALL POINTS

rotation

direct and either the identity or f fixes only 1 point called center of rotation (f(P)=P) → DIRECT and fixes 1 POINT

congruence

2 sets of points are congruent if there’s an isometry mapping points (written for segments- |AB|=|A’B’|

Triangle congruence→triangles ABC and A’B’C’ are congruent if…

the respective angles and side lengths must be congruent (ordering matters)

Direct isometrys

Rotations and translations

not direct isometrys

reflections

similar triangles

if respective angles are congruent (AAA implies this)

medians

line segments that connects vertex to mid point of the other side

medians intersection

intersect at centroid

centroid

constructed by medians that intersect at this point, trisects medians (G)

incenter

common point made by angle bisectors of a triangle (I), center of inscribed circle

angle bisectors

splits the angle in half, making congruent angles

circumcenter

a point made by 3 perpendicular bisectors of a triangles vertex (O), center is the center of the circle that circumscribes the triangle

perpendicular bisectors

a line that cuts a side of the triangle in half at a right angle

euler line

straight line that passes through several key points

points on euler line

the circumcenter (O), centroid (G), orthocenter (H)

orthocenter (H)

common made point made by altitudes intersecting, end point of euler line

altitudes

a straight line segment from a corner to the opposite side, meeting that side at a right angle

nine point circle

passes through specific points on a triangle

points that make up 9 point circle

all midpoints of sides, bases of altitudes, midpoints of AH, BH, CH (H is orthocenter)

where do medians intersect

centroid

where do angle bisectors intersect

intersect at incenter (I), center of inscribed circle

where do perpendicular bisectors intersect

intersect at circumcenter (O)

how many axioms of euclidean geometry

5

axiom 1 of euclidean geom

there is a unique line between any 2 points

axiom 2 of euclidean geom

any line segment can be extended indefinetly

axiom 3 of euclidean geom

a circle can be any center and radius

axiom 4 of euclidean geom

any 2 right angles are congruent

axiom 5 of euclidean geom

line L and point P not on line L, there’s a unique parallel line L2 thru P, does not interct line L

pythag theorem

a²+b²=c² → in a right triangel

Congruent triangle theroems

SSS (all sides =), ASA (2 angles and connected side), SAS (2 sides and angle between), SAA (2 angles not included on side)

Triangle theorems yeild…

CONGRUENCE

alternate interior angles

angles are on the same line, next to eachother, or congruent only for parallel lines

Triangle angle sums

180

Pons asinorum

2 base angles of an isosceles triangle are congruent

Star trek lemma

measure of an inscribed angle in a circle is half the angular measure of the arc it subtends to

angles that subtend the same arc…

are congruent (Bow tie lemma)

bow tie lemma/angles in same segment theorem

inscribed angles in a circle that subtend the same arc or chord are equal.

Ratio proporties

are for similar triangles →corresponding sides have to be equal |AB|/|A’B’| = |AC|/|A’C’| (can do cross multiplication)

power of the point

if 2 chords AB and CD intersect inside a circle at a point P, then PA(PB) = PC(PD)

law of cosines

c² = a² + b² - 2abCOS(C)

herons formula is used to…

find the area of a triangle when you only know all 3 side lengths

herons formula

square root ((s(s-a)(s-b)(s-c))

s in herons formula

semi-perimeter (half the perm), found by (a+b+c)/2

law of sines

a/SinA = b/SinB = c/SinC

rs formula is..

to find the area of a triangle using its inradius (the radius of its inscribed circle) and its semi-perimeter (half of its perimeter)

identity fixes how many points

all points

area of incircle

rs(use herons formula)pi r² → where r is the inradius (radius of incircle)

area of circumcircle

C=pi (2R)² → where 2R = a/sinA