Name: _________________________________________________________ 14-9 Circles Review Geometry Pd. ______ Date: __________

14-9: Circles Review

Circle Equations Summary Sheet Big Ideas 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 form 1. Move loose numbers to one side 2. Group x’s and y’s 3. 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 and 𝑦2

Systems with Circles Any point of intersection is a solution to the system – solve graphically- BRING COMPASS

Area of a Sector

𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑆𝑒𝑐𝑡𝑜𝑟 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝐶𝑖𝑟𝑐𝑙𝑒 (𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟

360)

Area of a circle = 𝜋𝑟2

Arc Length of a Sector (In degrees)

𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟

360)

Circumference of a circle = 𝜋𝑑

Solving for arc length IN RADIANS

s = r𝜃 where: s = arc length; r = radius; 𝜃 = central angle

Circle Theorems Summary Sheet

CENTRAL ANGLES VERTEX MUST BE ON THE CENTER OF THE

CIRCLE. RULE: ANGLE = INTERCEPTED

ARC

INSCRIBED ANGLES VERTEX 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 circle

OPPOSITE ANGLES ARE SUPPLEMETARY (ADD UP tO 180 )

SPECIAL INSCRIBED ANGLES Formed by a Tangent and chord

Rule: ANGLE =HALF THE INTERCEPTED

ARC

Sneaky Angle Formed 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−𝑵𝑨𝑹𝑪

𝟐

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.

4) Highlight parts you're solving for (in different colors).

5) Fill in/mark up the diagram.

Thales Theorem

Watch out! This is not

necessarily a parallelogram,

so consecutive angles are not

supplementary. ONLY

OPPOSITE ANGLES.

SPECIAL CASE If the chord is a

diameter, rt angles!

Special Case- “ Ice cream

Cone”

Narc + Outside angle = 180

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 tangent and diameter/radius are perpendicular at the point of tangency.

Tangents that meet at one exterior point are congruent:

Circle Proofs Tips

Mark up your diagram

Come up with a plan

Use your proof pieces

Look for congruent angles and congruent sides

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

3 common

tangents

4 common

tangents

part x part = part x part

or

pp = pp

Whole * outside = whole * outside

or

WO = WO

Fill in the following chart for each topic by placing a check mark in the box that describes your knowledge of each topic. ** Be honest! It’s just you looking at this! **

Topic This is easy… This is okay… This is really difficult …

1. Writing Circle Equations

2. Identifying Center-radius from circle equation

3. Completing the square to get equation of circle in center-radius form/Graphing

4. Writing Circle Equations

5. Writing Circle Equations

6. Graphing Circles

7. Systems of Equations

8. Arc Length and Sector Area Given Degrees

9. Sector Area

10. Arc Length (degrees)

11. Arc Length (radians)

12. Bowtie Angles/Working Backwards

13. Inscribed Angles and Diameters

14. Bowtie Angles

15. Special Inscribed Angles

16. Outside Angles/ Drawing diagrams

17. Outside Angles/Working Backwards

18. Angle/Segment Theorems in Circles

19. Angle/Segment Theorems in Circles

20. Segment lengths ( WO = WO)

21. Segment lengths ( WO = WO) *with factoring

22. Segment lengths ( PP = PP)

23. Common Tangents from a point

24. Angle/Segment Theorems in Circles

25. Drawing Common Tangents to two circles

26. Big Circles

27. Circle Proof

28. Circle Proof

29. Circle Proof