Incenter of a Triangle

One of the points of concurrency - it is the intersection point of the angle bisectors of the triangle

Circumcenter of a Triangle

One of the points of concurrency - it is the intersection point of the perpendicular bisectors of the sides of a triangle

(It is NOT necessarily in the triangle.)

Centroid of a Triangle

One of the points of concurrency - it is the intersection of the medians of the triangle

(In coordinate geometry, the coordinates of the centroid are the respective means of the coordinates of the vertices.)

Centroid Theorem

The distance from any vertex of the triangle to the centroid is 2/3 the length of the median

Incenter Theorem

The incenter of a triangle is equidistant from each side of the triangle (and therefore is the center of the inscribed circle).

Orthocenter of a Triangle

One of the points of concurrency - it is the intersection of the altitudes of a triangle

Euler Line

The line which passes through the orthocenter, circumcenter and centroid of a triangle.  (Those three are always collinear!)  

In what kind of triangle are the four points of concurrency the same point?

Equilateral triangles

The Area of a Regular Polygon

(Method 1)

A = 1/2(a)(P)

where a = apothem and P = perimeter

Area of a Regular Polygon

(Method 2)

Step 1: Draw triangles and find area of one  

A_T = 1/2 bc (sin A)  

Step 2: Multiply A_T by number of sides of polygon 

Names of polygons with 4, 5, 6, 7, 8, 9, 10 and 12 sides, respectively

quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, dodecagon

Surface Area of Prism

S.A. = Ph + 2B

(or combined areas of each face and both bases)

Surface Area of Cylinder

S.A. = 2(pi)r² + 2(pi)rh

Surface Area of a Pyramid

SA = 1/2 Pl + B

P - Perimeter

l - slant height

B - area of base

Surface Area of a Cone

SA = (pi)r² + (pi)rl

(Remember the cheer?)

Surface Area of a Sphere

S.A. = 4(pi)r²

Geometric Mean

If x is the geometric mean of a and b, 

then a/x = x/b

(or x² = ab)

Rhombus - definition

a parallelogram in which all four sides are congruent

Rectangle - definition

a parallelogram in which all four angles are congruent

square - definition

a parallelogram in which all angles and all sides are congruent 

(square = rhombus + rectangle)

Which quadrilaterals have congruent diagonals?

Squares, rectangles and isosceles trapezoids 

(those that can stand "upright")

Which parallelograms have diagonals that bisect each other?

all of them

Parallelogram - definition

a quadrilateral in which opposite sides are parallel

Five ways to prove that a quadrilateral is a parallelogram

1.  Both pairs of opposite sides parallel

2.  Both pairs of opposite sides congruent

3.  One pair of opposite sides parallel and congruent

4.  Consecutive angles supplementary (overlapping pairs)

5.  Diagonals bisect each other

[image]

ab = cd

m<C = 1/2 m(arc AB)

m<M = m(arc AB)

adjacent (same vertex, common ray)

Linear pair (adjacent and formed by opposite rays)

Corresponding angles

(Same positions relative to the "parallel" lines and the transversal.  Of course, the lines are not necessarily parallel.)

Alternate interior angles 

(between the two "parallels" and on opposite sides of the transversal)

vertical angles (always congruent)

Name of highlighted angle

and relationship to angles in triangle[image]

"Exterior angle" of the triangle

1.  supplementary to <CBA

2.  equal to the sum of the two remote interior angles (<A and <C)

secant line - intersects the circle twice

Sketch a chord, tangent line and secant line

("The whole times the outside = the whole times the outside.)

x² = 5(12)

x = 2√15

1.  BD is the geometric mean of AD and DC.

2.  AB is the geometric mean of AC and AD.

3.  BC is the geometric mean of AC and CD.

The sum of the interior angles of a pentagon

3 (180) = 540 (degrees)

You can draw triangles from ONE vertex and count the "180's" or you can start with one exterior angle and work your way in.

Formula:  S = (n-2)180

Area of an Equilateral Triangle

A = s² √(3) / 4

Area of Kites and Rhombi

A = 1/2 d₁d₂

The slope of the line passing through

(f,p) and (n,s)

slope = (p-s)/(f-n)

Angle of Depression

Angle of Elevation

Ratios of the sides of a 30-60-90

Ratios of the sides of a 45-45-90

m(<P) + m(arc AB) = 180

(You could go the long way and subtract the minor arc from the major arc and divide by 2.)

The Triangle Inequality Theorem

The sum of any two sides is greater than the third.

(If given three sides, put them in ascending order and the first two must add up to be greater than the third.)

The Equidistance Theorems

1.  If a point is equidistant to the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

2.  If two points are both equidistant to the endpoints of a segment, then they determine the perpendicular bisector of the segment.

3.  If a point is equidistant to the sides of angle, then it lies on the angle bisector.  

Measure of <ABC [image]

1/2 m(arc AB)

Point-slope form of a line

y - y₁ = m(x - x₁)

Slopes of parallel and perpendicular lines

Parallel lines have same slope.

Perpendicular lines have slopes that are opposite reciprocals of each other.

Side-Splitter Theorem

AB/AC = BD/CE

[image]

Midsegment of a Triangle

The segments which join the midpoints of any two sides of a triangle.

1.  Each midsegment is parallel to the third side.

2. Its length is 1/2 the length of the third side.

m(<P) = 1/2 [m(arcAC) - m(arcBD)]

When do you use Heron's Formula?

To find the area of a triangle, given SSS

Heron's Formula

(The square root symbol here is a bit short.  It should cover all four factors.)

√s(s-a)(s-b)(s-c)

where s = the semi-perimeter and a,b,c are the sides.

5/3.5 = 12/x

x=42/5

What is the only segment inside a triangle that will always allow us to set up a proportion, no matter what kind of triangle?

angle bisector

Given any three points, how many planes pass through them?

Trick question...

1 plane if points are noncollinear

Infinite number of planes if the points are collinear (lie on one line)

Name the situations with points and/or lines which guarantee one (and only one) plane.

1.  two parallel lines

2.  a line and a point not on it

3.  three non-collinear points

4.  two intersecting lines