Visuals Chapter 6 Tim Russell. Types of Visuals Nonprojected Visuals Projected Visuals.
FULL YEAR of High School GEOMETRY GEOMETRY ......FULL YEAR of High School GEOMETRY GEOMETRY FORMULA...
Transcript of FULL YEAR of High School GEOMETRY GEOMETRY ......FULL YEAR of High School GEOMETRY GEOMETRY FORMULA...
FULL YEAR of High School GEOMETRY
GEOMETRY FORMULA &
REFERENCE PACKET7 pages of formulas, theorems, visuals and tricks! Must-have for students preparing for
end of course exams!
Created by: KoltyMath
GEOMETRY REFERENCE SHEET
Coordinate GeometryDistance Formula/LengthD = x2 − x1
2 + (y2 − y1)2
Midpoint Formula
Partitioning formulaPx = x1 + k(x2 – x1)
Py = y1 + k(y2 – y1)
Ratio a:b
= (x1+x2
2,
y1+y2
2)xm, ym
Finding the endpointPythagorean theorem
can replace the
distance formula.
a2 + b2 = c2
(–4, 2)
(1,–5)
(6,–12)
+5 –7
+5 –7
Visual Method
𝟒
𝟏𝟎=
𝟔
𝟏𝟓=
𝟐
𝟓=
𝐀𝐏
𝐀𝐁
Part
Whole
k
Rati
o AP
:PB
= 2:
3
Slope Formula
?
m =y2 − y1
x2 − x1=
rise
run
m = 2
Types of Slope
Linear Equations
y = mx + b
y − y1 = m(x − x1)
Slope-Intercept Form
Point-Slope Form
Mr.
Slo
pe G
uy
Negative – Positive + Undefined Zer0x = –2 y = 2
b
y = – ½ x + 2
KoltyMath
Angle RelationshipsParallel lines cut by a transversal
acute = acute
obtuse = obtuse
acute + obtuse = 180°
Alternate Interior
Angles Theorem
Consecutive Interior Angles Theorem
“Same Side Interior” Angles Theorem
Angle Types
Complementary = 90°
Supplementary = 180°
TrianglesTriangle Sum Theorem
a + b + c = 180
Exterior Angle Theorema + b = external c
Side-Angle RelationshipsIf a > b > c (sides)
Then A > B > C (angles)
Triangle Possible side lengths
Sum (+) of 2 shorter sides > 3rd side
Triangle Inequality Theorem
Range of Possible Values for a 3rd Side
Sum > x > Difference+ –
Ex: 7, 4, x
11 > x > 3
Triangle Types
60 60
60
Scalene Isosceles EquilateralEquiangular
Right
Angle Classification:
acute, obtuse, right, equiangular
Side Classification:
scalene, isosceles, equilateral
Isosceles Triangle Theorems Hinge Theoremaka SAS Inequality: the measure of the included angle between two pairs of
congruent sides dictates which triangle has the longer third side.
CD > AB
Altitude Theorem
o Bisects Vertex Angle
o Bisects Base Side
KoltyMath
CONGRUENCEASA
Angle-Side-Angle
AAS
Angle-Angle-SideHL
Hypotenuse-Leg
SSS
Side-Side-Side
SAS
Side-Angle-Side
Once the▲s are proven
congruent you can use CPCTC.
Similarity (Match it up!)Triangle similarity proofs
AA~ SAS~ SSS~
Angles must be Congruent and
Sides must be in Proportion
Similarity Tips
✓ Match! Match! Match!
✓ Proportions
Side Splitter TheoremIf a parallel segment intersects 2 sides of a triangle.
U1
U2=
L1
L2
U1
W1=
U2
W2
Similar Polygons
Sca
le F
acto
r Side Lengths a : b
Perimeter a : b
Area a2 : b2 (surface area too)Volume a3 : b3 (similar solids)
Using the appropriate scale factor
allows you to find missing
perimeters/areas/volumes with limited
information by setting up proportions.
a2
b2 =AreaA
AreaB
Ex:
Mean Proportional TheoremsUsed when an altitude is drawn from the
right angle to the hypotenuse of the▲.
Altitude Theorem Leg Theorem
short
altitude=
altitude
long
long
altitude=
altitude
short
Whole
Leg 1=
Leg 1
Part 1
Whole
Leg 2=
Leg 2
Part 2
KoltyMath
TransformationsTranslations: “shift”(x,y) (x + a, y + b)
+– +
–
Reflections: “FLIP”(x,y) ______
Ry-axis (–x,y)
Rx-axis (x,–y)
Ry=x (y,x)
Ry=-x (–y,–x)
You can always use a visual counting method instead of
using rules for reflections.
3
3
Orie
ntat
ion
Chan
ges
afte
r a
Refle
ctio
n
Rotations: “turn”(x,y) ______ (centered at origin)
Clockwise Counterclockwise
– +
R90/-270 (–y,x)
R180/-180 (–x,–y)
R270/-90 (y,–x)
When you connect the pre-image and image to the
center of rotation, you form the degree of rotation.
Remember: you can also perform rotations
centered at the origin by turning your paper.
Dilations: “Grow/Shrink”(x,y) (kx, ky) (centered at origin)
Only Non-Rigid Transformation
Scale Factor (k) =new
oldOnly non-rigid transformation. Dilations enlarge or
shrink your object and form SIMILAR FIGURES.
Image, Pre-image and Center of Dilation are collinear.
To perform dilations not centered at the origin, you can
plot the center and pre-image points and use rise and
run to find the image points. Option 2, use the formula:
x’ = a + k(x – a) y’ = b + k(y – b)
k = 2
Right Triangles
a2 + b2 = c2
Pythagorean Theorem
Converse of the Pythagorean theoremTo determine if the triangle is acute, right or obtuse
from the side lengths only. Center of Dilation: (a,b)Pre-image Point: (x,y)Image Point: (x’,y’)
Notice: when you connect
the pre-image and image
points with a line, the point
of intersection for all three
lines is the center of
dilation (0,-2).
KoltyMath
trigonometrySimplifying radicals
Perfect Squares
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
20
4 5
2 5
Grea
test
Per
fect
Squ
are
= 12 25 2
4 5 ● 3 10
= 12 50
= 12●5 2
= 60 2
Special right trianglesAll 30-60-90 and 45-45-90 are similar and their sides can be found using the side relationships below.
30-60-90 45-45-90Isosceles Right Triangle
Useful with equilateral triangles when an altitude is drawn in.
Useful with squares when a diagonal is drawn in.
TrigonometrySOH CAH TOA
𝐭𝐚𝐧 𝟒𝟎 =𝟏𝟎
𝐱
x =10
tan 40
x = 11.92
𝐬𝐢𝐧Θ =𝟑
𝟓= . 𝟔
Θ = sin−1(.6)
Θ = 36.87°
If you’re solving for the angle (Θ)
Use the INVERSE TRIG Functions-1
Θ = 𝐬𝐢𝐧−𝟏𝐨
𝐡
Θ = 𝐜𝐨𝐬−𝟏(𝐚
𝐡)
Θ = 𝐭𝐚𝐧−𝟏(𝐨
𝐚)
Complementary angles
Sin A = Cos B
Sin B = Cos A
Ex 1: Sin T = Cos (90 – T)
Ex 2: Sin 30 = Cos 60
Special segments & CentersCircumcenter
formed by
perpendicular bisectors
result: AM=BM=CM
Incenter
formed by
angle bisectors
result: XM=YM=ZM
Centroid
formed by Medians
divides median into a 2:1BM = 2(MZ)
Orthocenter
formed by Altitudes2
(x1+x2+x3
3,
y1+y2+y3
3)
KoltyMath
Spatial reasoningArea Formulas (2d Shapes)Circle A = Πr2
C = 2Πr
Triangle A = ½ bh
Parallelogram A = bh
Rectangle A = bh
Rhombus/Kite/Square A = ½ d1 d2
Square A = s2
Trapezoid A = ½ h(b1+b2)
Any Regular Polygon A = ½ aP
a = apothemP = perimeter
n = number of sides
may
nee
d tr
ig
to
sol
ve fo
r bo
th
central angle = 𝟑𝟔𝟎
𝐧
Volume & Surface area formulas
Prism/Cylinder 𝐕 = 𝐁𝐡
Pyramid/Cone 𝐕 =𝟏
𝟑𝐁𝐡
Sphere 𝐕 =𝟒
𝟑Π𝐫𝟑
VOLUME FORMULAS
SURFACE AREA FORMULAS
Prism 𝐋𝐀 = 𝐏𝐡𝐒𝐀 = 𝐏𝐡 + 𝟐𝐁
Cylinder 𝐋𝐀 = 𝟐Π𝐫𝐡𝐒𝐀 = 𝟐Π𝐫𝐡 + 𝟐Π𝐫𝟐
Pyramid 𝐋𝐀 =𝟏
𝟐𝐏𝐥
𝐒𝐀 =𝟏
𝟐𝐏𝐥 + 𝐁
Cone 𝐋𝐀 = Π𝐫𝐥𝐒𝐀 = Π𝐫𝐥 + Π𝐫𝟐
Sphere 𝐒𝐀 = 𝟒Π𝐫𝟐
SA = LA + 2B
l = slant length
h = height
r = radius
B = Area of Base
P = Perimeter
12 in = h
13 in = l
5 in = a
circlesAngles in a circle
Central ∠ Inscribed ∠ Inscribed Right ∠
Intercepted Arc Tangent Radius Supplementary Opposite ∠s
ALTERNATE OPTION
Break the solid into its net
and find the area of each
face separately then add
them all together.
x = ½ (arc 1 + arc 2)
a●b = x●yIntersecting Chords
ext.(whole) = ext.(whole)Secant-Secant Rule
ext.(whole) = tangent2
Secant-Tangent Rule
Segment lengths in a circle
x = ½ (far arc – near arc)
Sector Area & arc length
Sector Area
Πr2 =Central Angle
360
Arc Length
2Πr=
Central Angle
360
Area of Circle
Circumference
Θ
Θ
Θ
Θ
Lateral Area (LA)is the
area of all the surfaces
EXCEPT for the BASE.
KoltyMath
Angles of a polygon
Density 𝐃 =𝐌𝐚𝐬𝐬
𝐕𝐨𝐥𝐮𝐦𝐞
𝐏𝐨𝐩𝐮𝐥𝐚𝐭𝐢𝐨𝐧 𝐃𝐞𝐧𝐬𝐢𝐭𝐲 =𝐏𝐨𝐩𝐮𝐥𝐚𝐭𝐢𝐨𝐧
𝐋𝐚𝐧𝐝 𝐀𝐫𝐞𝐚
Degree & radian conversion
𝐑●180
Π
Radians to Degrees
𝐃●Π
180
Degrees to Radians
D =
Degr
ees
R =
Radi
ans
Equation of a circle(𝐱 − 𝐡)𝟐+(𝐲 − 𝐤)𝟐= 𝐫𝟐
(𝐱 − 𝟑)𝟐+(𝐲 + 𝟓)𝟐= 𝟏𝟒𝟒Ex:
Center: (3,–5) r = 12
Quadrilateral properties
Quadrilateral
360°
1. Opposite sides are parallel (II)
2. Opposite sides are congruent (≌)
3. Opposite angles are congruent (≌)
4. Consecutive angles are supplementary
5. Diagonals bisect each other (same midpoint)
Parallelogram
Rectangle
1. 4 right angles
2. Diagonals are congruent (≌)
Rhombus
1. 4 congruent (≌) sides
2. Diagonals are perpendicular ()
3. Diagonals bisect the angles
Square
Absorbs all properties from the
parallelogram, rectangle, and rhombus
Trapezoid
1. Only 1 pair of parallel (II) sides
2. Same Side Interior Angles are Supplementary
3. Median = ½ (Base 1 + Base 2)
Isosceles Trapezoid
1. Legs are congruent (≌)
2. Base angles are congruent (≌)
3. Diagonals are congruent (≌)
1. 2 Pairs of Consecutive Sides are congruent
2. Diagonal BD bisects ∠B and ∠D
3. Diagonals are perpendicular
4. ≌ opposite angles formed at ∠C and ∠A
KiteB D
A
C
KoltyMath
A = ½ d1d2
A = ½ d1d2
A = ½ d1d2
A = ½ d1d2Area = ½ diagonal 1 * diagonal 2