Geometry Final Formulas and Vocabulary

Plane

3 noncollinear points

Intersection of 2 planes

Line

Coplanar

Lie in same plane

If a plane intersects 2 parallel lines, then the intersection is

2 parallel lines

2 lines perpendicular to the same plane

Coplanar

If line l is parallel to plane P, how many planes containing line l can be drawn parallel to plane P

1

2 lines intersect at a

point

In space, how many planes can be perpendicular to a given line at a given point on that line in space

1

Through a given point, how many lines can be drawn perpendicular to a given plane

1

Angle Bisector

Ray that bisects an angle by dividing it into 2 congruent angles

Angle addition postulate

If S is in the interior of <QRT, then m<QRS+m<SRT=m<QRT

Reflexive Property of Congruence

AB ≅ AB

Symmetric Property of Congruence

If AB ≅ CD, then CD ≅ AB.

Transitive Property of Congruence

If AB ≅ CD and CD ≅ EF, then AB ≅ EF.

Segment Addition Postulate

If B is between A and C, then AB + BC = AC

Triangle Congruency

SSS, SAS, ASA, AAS, HL

CPCTC

Corresponding parts of congruent triangles are congruent

Perpendicular Bisector Theorem

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

Angle Bisector Theorem

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

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Midpoint Formula

(x₁+x₂)/2, (y₁+y₂)/2

Circumcenter of a Triangle

The point where the three perpendicular bisectors of a triangle meet

Incenter of a Triangle

The point where the three angle bisectors meet

Inscribed Circle

Largest circle that fits inside a triangle

Incenter

center of inscribed circle

Circumscribing Circle

Smallest circle that contains the whole triangle

Circumcenter

Center of the circumscribing circle

Median

line segment that joins the vertex to the midpoint of the opposite side

Centroid

The point where the three medians of a triangle meet

Altitude

A perpendicular segment that connects a side to the opposite vertex

Orthocenter

The point where the three altitudes meet

Convex polygon

All interior angles are less than 180°.

Concave polygon

One or more of the interior angles is more than 180 degrees

Polygon angle sum theorem

180(n-2)

Rhombus

A parallelogram with four equal sides and diagonals bisect at right angles

Kite

2 pairs of adjacent sides of equal length

Positive rotation angle

counterclockwise

Negative Rotation Angle

clockwise

Reflection across y-axis

(x,y) -> (-x,y)

Reflection across x-axis

(x,y) -> (x,-y)

Reflection across x=y

(x,y) -> (y,x)

Rotation 90 degrees

(x,y) -> (-y,x)

Rotation 180 degrees

(x,y) -> (-x,-y)

Rotation 270 degrees

(x,y) -> (y,-x)

Rotation 360 degrees

(x,y) -> (x,y)

Triangle Similarity

AA, SSS, SAS

Pythagorean Theorem

a² + b² = c²

Acute triangle

c²<a²+b²

Obtuse Triangle

c²>a²+b²

Diagonal Formula

y=√(v^2+w^2+x^2)

45-45-90 triangle

x, x, x√2

30-60-90 triangle

x, x√3, 2x

Sine of x°

sin x=cos (90-x)°

Cosine of x°

sin(90-x)°=cos x°

Law of Sines

a/sinA = b/sinB = c/sinC

Law of Cosines

a²=b²+c²-2bcCosA

Rearranged Law of Cosines

cosA=b^2+c^2-a^2/2bc

Area of a triangle

A= 1/2 bh

Area of any triangle

1/2absinC

Heron's Formula

A=√s(s-a)(s-b)(s-c)

Semiperimeter of a triangle

s= a+b+c/2

Area of a parallelogram

A=bh

Area of a trapezoid

A=1/2(a+b)h

Distance Between Points Formula

d=√((x2 - x1)² + (y2 - y1)²)

Secant

a line that intersects a circle at 2 points

Chord

A line segment with both endpoints on the circle

Tangent

A line that intersects the circle in exactly one point

Concentric Circles

coplanar circles with the same center

Tangent circles

two coplanar circles that intersect at exactly one point

Equation of a circle centered at the origin

x² + y² = r²

Equation of circle not centered at origin

(x-h)^2 + (y-k)^2 = r^2

Circumfrence formula

C=2πr

Area of circle

A = πr²

Area of sector AOB

πr^2θ/360

Arc Length Formula

πrθ/180

Arc length in Radians

s = θr

Sector Area in Radians

S= θr^2/2

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc

2 secants intersect outside circle

b(a+b)=c(c+d)

A secant and tangent intersect outside a circle

a^2=d(c+d)

2 chords intersect outside a circle

HF(FS)=TF(FP)

2 chords intersect inside a circle

m<1=1/2(AB+CD)

Tangent and chord intersect on the circle

m<1=1/2 (BAC)

2 tangents intersect outside the circle. What is the angle formula

m<1=1/2 (ACB-AB)°

2 secants intersect outside the circle. What is the angle formula

m<1= 1/2 (AB-DC)°

A tangent and a secant intersect outside the circle. What is the angle formula

m<1= 1/2(AC-DB)°

Volume of cube

V=s³

Volume of a rectangular prism

V = lwh

Surface area of a rectangular prism

SA=2lw+2lh+2wh

Lateral Surface Area of a Rectangular Prism

LA=2wh+2lh

Surface area of a cube

SA=6s^2

Length around edges of rectangular prism

4(l+w+h)

Lateral surface area of cube

LA = 4s²

Length around edges of cube

12l

Lateral area of prism

LA=Ph

Surface area of prism

SA=Ph+2B

Volume of a pyramid

V=1/3Bh

Lateral Area of a Pyramid

LA=1/2pl

Surface Area of a Pyramid

SA= 1/2Pl + B

Volume of cylinder

V = πr²h