Circles - Cheat Sheet Topic Overview

Center Radius Form, for circle centered at (h,k) with radius, r(x−h)2+( y−k )2=r2

Completing the square for center-radius form1. Move loose numbers to one side

2. Group x’s and y’s3. Divide middle term by 2 and square it – ADD TO BOTH SIDES!

4. put factors into Squared Form ()2 ( remember the number will be half of the middle term)5. You’re in center-radius form!!!

We complete the square twice to put general form equations of circles into Center-Radius form, then graph! Recognize a circle by finding an x2

Systems with CirclesAny point of intersection is a solution to the system – solve graphically!

Systems with ParabolasAny point of intersection is a solution to the system – solve graphically!

Watch out for sneaky turning points and sneaky solutions- know how to manipulate your calculator!( 12-6 was a quiz!)Area of a Sector

Areaof a Sector=Area of a˚

( angle measure of sector360 )

Area of a circle = π r2

Arc Length of a Sector (In degrees)

Arc Length=Circumference( angle measure of sector360 )

Circumference of a circle = πd

Solving for arc length IN RADIANSs = rθ where: s = arc length; r = radius; θ = central angle

Radians – unit of angle measureAn angle is 1 radian when the length of the arc of the circle is equal to the radius

ConversionsSet up a proportion and solve for desired angle measure!

radiansdegrees

= π180

Formulas for area and circumference are on the reference sheet.

KNOW DIFFERENCE B/T AREA and LENGTH (CIRCUMFERENCE)

Part 1- Circles

Proving Circles Similar

1. Translate to get centers to coincide2. Dilate with center of dilation at one of the centers and scale factor using

the radii.Circle Regents questions

1. The equation of a circle is . What are the coordinates of the center and the length of the radius of the circle?1) center and radius 42) center and radius 43) center and radius 164) center and radius 16

2. A circle with a radius of 5 was divided into 24 congruent sectors. The sectors were then rearranged, as shown in the diagram below.

To the nearest integer, the value of x is1) 312) 163) 124) 10

3. In the diagram below of circle O, the area of the shaded sector AOC is and the length of is 6 inches. Determine and state .

4. A designer needs to create perfectly circular necklaces. The necklaces each need to have a radius of 10 cm. What is the largest number of necklaces that can be made from 1000 cm of wire?1) 15

2) 163) 314) 325.

YOU TRY!

6. If is the equation of a circle, the length of the radius is1) 252) 163) 54) 4

7. Triangle FGH is inscribed in circle O, the length of radius is 6, and .

What is the area of the sector formed by angle FOH?1)

2)

3)4)

Circle THEOREMS - Cheat SheetCENTRAL ANGLES

VERTEX MUST BE ON THE CENTER OF THE CIRCLE.

RULE: ANGLE = INTERCEPTED ARC

INSCRIBED ANGLESVERTEX MUST BE ON THE CIRCLE.

RULE: ANGLE =HALF THE INTERCEPTED ARC

ANGLES FORMED BY 2 CHORDS

“BOW –TIE ANGLES”VERTEX is NOT on the center and

NOT on the circle.

RULE: ARC¿1+ ARC ¿2 ¿

2

Cyclic Quadrilaterals

Quadrilateral inscribed in a circleOPPOSITE ANGLES ARE

SUPPLEMETARY (ADD UP tO 180 )

SPECIAL INSCRIBED ANGLESFormed by a Tangent and chord

Rule: ANGLE =HALF THE INTERCEPTED ARC

Sneaky AngleFormed by a secant and chord

Rule: Find the measure of inscribed adjacent angle and subtract from 180.

ANGLES FORMED BY TWO SECANTS, OR TWO TANGENTS OR A SECANT AND A TANGENT.

RULE: Outside ∡=FARC−NARC2

Thales Theorem

Watch out! This is not necessarily a parallelogram, so consecutive angles are not supplementary. ONLY OPPOSITE ANGLES.

SPECIAL CASEIf the chord is a

diameter, rt angles!

Special Case- “ Ice cream Cone”Narc + Outside angle = 180

1) Always solve for arcs first. (HIGLIGHT DIAMETERS!).KNOW HOW TO WORK WITH RATIOS!2) Solve the parts in the order they are presented in for the problem.3) GO SLOW.

Parallel Chords

The arcs between two parallel chords are congruent

Congruent Chords

The arcs outside (subtended by) congruent chords are congruent

Segment Length INSIDE Circle (PP)

If a diameter/radius is perpendicular to a chord, then it

bisects that chord.

Segment Length

OUTSIDE Circle

A right angle is formed at the point of tangency between a tangent and diameter/radius

Tangents that meet at one

exterior point are congruent:

Big Circles (Yesterday’s Review)!Similar Circles1. Translate (if necessary) first - STATE VECTOR2. Dilate at a center of dilation and a scale factor (image / pre-image)3. ConcludeCircle Proofs Tips

Mark up your diagram Come up with a plan Use your proof pieces

part x part = part x partor

pp = pp

3 common tangents

4 common tangents

whole x outer = whole x outer

orwo = wo

Look for congruent angles and congruent sides Always look for radii (always congruent) and Inscribed Angles!!!

Circle Regents questions

1. In circle O shown below, diameter is perpendicular to at point C, and chords , , , and

are drawn.

Which statement is not always true?1)

2)

3)

4)

2. In the diagram shown below, is tangent to circle O at A and to circle P at C, intersects at B, , , and

.

What is the length of ?1) 6.42) 83) 12.54) 16

3. In the diagram below, quadrilateral ABCD is inscribed in circle P.

What is ?1) 70°2) 72°3) 108°4) 110°

You try!

4. In the diagram below, , , , , and are chords of circle O, is tangent at point D, and radius is drawn. Sam decides to apply this theorem to the diagram: “An angle inscribed in a semi-circle is a right angle.”

Which angle is Sam referring to?1)2)3)4)

5. In the diagram below of circle O with diameter and radius , chord is parallel to chord .

If , determine and state .

6. Parallel secants and intersect circle O, as shown in the diagram below.

If and , then equals

1) 1062) 1153) 1304) 156Part 2- Quadrilaterals

Quadrilateral Properties Cheat Sheet – Sides and DiagonalsParallelogram Properties

2 sets of opposite sides are parallel2 sets of opposite sides are congruentDiagonals bisect each otherOpposite angles are congruent, Consecutive angles are supplementary

Rectangle Properties2 sets of opposite sides are parallel2 sets of opposite sides are congruentDiagonals bisect each other*Diagonals are congruent*Adjacent sides are perpendicular

Rhombus Properties

2 sets of opposite sides are parallel2 sets of opposite sides are congruentDiagonals bisect each other*All sides are congruent*Diagonals are perpendicularDiagonals bisect vertex angles

Square Properties2 sets of opposite sides are parallel2 sets of opposite sides are congruentDiagonals bisect each otherDiagonals are congruentAdjacent sides are perpendicularAll sides are congruentDiagonals are perpendicular

Trapezoid PropertiesONLY 1 set of opposite sides are parallelAngles between bases are supplentary

Isosceles Trapezoid PropertiesONLY 1 set of opposite sides are parallel*ONLY 1 set of opposite sides are congruent (legs)*Diagonals are congruent, BASE ANGLES ARE CONGRUENT

Tools:Slope: Midpoint: Distance:

d= √(x2−x1)2+( y2− y1)

2

Helps prove:

m= y2− y1

x2−x1

or counting method

M=(x1+x2

2,

y1+ y2

2)

Quadrilateral Family “Geome-tree”

Another way to think about it….

How you read this: All rectangles, rhombuses, and squares are parallelograms, but not all parallelograms are rectangles, squares and/or rhombuses. All squares are rectangles but not all rectangles are squares etc….

Types of Quadrilateral Regents Questions:

Helps prove:Helps prove:

o Property Multiple Choice Questionso Using a property to solveo Coordinate Geometry Proofo Two-Column Quadrilateral Proof

Quadrilateral Regents Questions

1. Quadrilateral ABCD has diagonals and . Which information is not sufficient to prove ABCD is a parallelogram?1) and bisect each other.2) and 3) and 4) and

2. The diagram below shows parallelogram LMNO with diagonal , , and .

Explain why is 40 degrees.

3. In the coordinate plane, the vertices of are , , and . Prove that is a right triangle. State the coordinates of point P such

that quadrilateral RSTP is a rectangle. Prove that your quadrilateral RSTP is a rectangle. [The use of the set of axes below is optional.]

You Try!

4. Which statement is not always true about a rhombus? 1)

The diagonals are perpendicular

2)

The opposite sides are congruent.

3)

The adjacent sides are perpendicular.

4)

The opposite sides are parallel.

5. In the accompanying diagram of rhombus ABCD, Sides A B and BC are adjacent sides, If AB=4 x , and BC=x+3

What is the value of x? What is the length of AB?

6. In the diagram below, quadrilateral STAR is a rhombus with diagonals and intersecting at E. , , , , ,

, and .

a) Solve for SR

b) Solve for RT

c) Solve for

Part 3- SolidsKEY CONCEPTS IMPORTANT NOTES

CALCULATING AREA OF REGULAR POLYGONS

Need to “break up the figure” into triangles

Steps:1. Calculate the apothem!2. If not already there, draw in the Apothem

(mark the right angles) and bisect the central angleto find the vertex angle of the small RIGHT triangle.

3. Bisect the base of the isosceles triangle to find the length of one side of the right triangle.

4. Use SOH CAH TOA to calculate the length of the apothem.5. Find the area of each triangle6. Multiply the area by the # of triangles in the regular polygon.

TRANSLATIONS FORMING SOLIDS, PROPERTIES AND CROSS SECTIONS

Cylinders Formed by translation Base shape: circle (parallel and congruent shapes in the 3D figure) Lateral View: Rectangle (cross section perpendicular to the base) Base View: Circle (cross section parallel to the base)

Prisms Formed by translation Polygonal bases Named by their base shapes Lateral View: Rectangle (cross section perpendicular to the base) Base View: same as base shape (cross section parallel to the base)

Pyramid Formed by translation and dilation One base that is a polygon Named by base shape Lateral edges congruent

ROTATIONS FORMING SOLIDS, PROPERTIES

AND CROSS SECTIONS

*Cylinders can be formed by rotations

Cone Formed by rotation of triangle Slant height: height from edge of base to top Base shape: circle (parallel and congruent shapes in the 3D figure) Lateral View: Triangle (cross section perpendicular to the base) Base View: Circle (cross section parallel to the base)

Sphere Formed by rotation of circle or semi-cirlce Lateral View: Circle (cross section perpendicular to the base) Base View: Circle (cross section parallel to the base) The great circle: largest circle within a sphere ; same diameter as sphere

VOLUME OF PRISMS AND CYLINDERS,

EQUAL VOLUMES, CAVALIERI’S PRINCIPLE

Volume of Prisms and Cylinders: V = Bh

(B = area of the base; h is the height/depth of the prism/distance between the bases)

Cavalieri’s Principle: two of the same solid that have thesame base area and the same height, also have the same volume.

VOLUME OF PYRAMIDS, CONES

AND SPHERES

V(pyramid/cone) = 13

Bh

V(sphere) = 43

πr3

DENSITY AND SURFACE AREA

Population Density A ratio of the amount of a population that exists over a given area

Population Density = populationtotal area

Density of a 3D Solid A ratio that compares an object’s weight (mass) to the amount of space (volume) it takes up

Density = Mass

Volume3 DIMENSIONAL SOLIDS PROPERTIES

Features: Prism Cylinder Cone Pyramids SphereBase View Always

polygon Circle Circle Any Polygon Circle

Lateral View Rectangle Rectangle Isosceles Triangle

Isosceles Triangle Circle

FormationTranslating a polygon into

3 Dimensions

Translating a circle into 3 Dimensions

OrRotating a Rectangle

Rotating a Triangle

Translation and Dilation of

a polygon

Rotate a semi-circle

Cross Section Parallel to the Base

Same 2D shape as base view

Cross Section Perpendicular to the Base

Same 2D shape as lateral view

Number of Bases

2 2 1 1 n/a

Example

Formulas (Volume)

V = (area of the base) height

V = (area of the base) height V =

13

(area of the

base) height

V =13

(area of the

base) height

Helpful Tips For Solids Be prepared for application problems Careful typing into your calculator (especially fractions) Don’t Round until the end of the problem Leave in terms of Pi until the end of the problem (sometimes pi will

cancel) Use appropriate formulas If you see equal height or equal volume, that’s Cavalieri’s Principle! Be prepared to describe how solids are formed through transformation Be prepared to describe what cross sections look like

Solids Regents Questions

1. Which object is formed when right triangle RST shown below is rotated around leg ?

1) a pyramid with a square base2) an isosceles triangle3) a right triangle4) a cone

2. The Great Pyramid of Giza was constructed as a regular pyramid with a square base. It was built with an approximate volume of 2,592,276 cubic meters and a height of 146.5 meters. What was the length of one side of its base, to the nearest meter?1) 732) 773) 1334) 230

3. Walter wants to make 100 candles in the shape of a cone for his new candle business. The mold shown below will be used to make the candles. Each mold will have a height of 8 inches and a diameter of 3 inches. To the nearest cubic inch, what will be the total volume of 100 candles?

Walter goes to a hobby store to buy the wax for his candles. The wax costs $0.10 per ounce. If the weight of the wax is 0.52 ounce per cubic inch, how much will it cost Walter to buy the wax for 100 candles? If Walter spent a total of $37.83 for the molds and charges $1.95 for each candle, what is Walter's profit after selling 100 candles?

You try!

4. If the rectangle below is continuously rotated about side w, which solid figure is formed?

1) pyramid2) rectangular prism3) cone4) Cylinder

5. Molly wishes to make a lawn ornament in the form of a solid sphere. The clay being used to make the sphere weighs .075 pound per cubic inch. If the sphere's radius is 4 inches, what is the weight of the sphere, to the nearest pound?1) 342) 203) 154) 4

6. Two containers show below hold candies of the same size. Container A holds 75 candies and container B holds 160 candies. Given the dimensions below, which container has a smaller population density?

7. A contractor needs to purchase 500 bricks. The dimensions of each brick are 5.1 cm by 10.2 cm by 20.3 cm, and the density of each brick is . The maximum capacity of the contractor’s trailer is 900 kg. Can the trailer hold the weight of 500 bricks? Justify your answer.