Topic 6 Test: Circles30 questions1. Definitions related to circles (Similar circles);and Derivation of Formulas (Area of a circle, Area of a sector, Circumference, Arc Length, etc. ): 9 questions2. Arcs and Angles, applying theorems related to arcs and angles( Find measure of Angles, etc. ): 7 questions3. Area of a Sector: 1 question4. Circles and Pythagorean Theorem: 1 question5. Arc Length: 1 question6. Equations of Circles: 6 questions7. Constructions: 4 questionsv Circle inscribed in trianglev Square inscribed in circlev Equilateral triangle inscribed in circlev Regular hexagon inscribed in circle

9.5: Distance in Coordinate Geometry (Equation of Circles)

Remember

This is the equation of a circle withcenter at the origin and radius of 8 units

But, not all circles are centered at the origin:

This is the equation of a circle centered at C (h,k) with a radius of r

Match the circle with its equation in vertex form

This is an example of the equation of a Circle in General Form

This is the equation of the same circle in standard( vertex form)

Standard form or vertex formCenter (-7,4) ; radius 8

Your TurnWrite the equation of the circle in standard form

Write the equation of the circle

Center(h,k)

(� − �)�+ (� − �)�= ��

(� − �)�+ (� − �)�= ��

To find the radiusUse Pythagorean Theorem�� + ��= ��

�� + ��= ��

9 + 9 = ��

18 = ��

c = �� = � �

(� − �)�+ (� − �)�= (� �)�

What is the center and radius of the circlegiven by the equation:�� + �� – �� + �� – 3 = 0

�� + �� – �� + �� – 3 = 0 Rearrange terms��– �� + �� + �� – 3 = 0 Complete square��– ��(+ �) + �� + �� (+ �)– 3 = 0 + 9 + 4Factor the perfect square and move -3 to other side of equation(� − �)�+ (� + �)�= 13 + 3(� − �)�+ (� + �)�= 16

Center (3, -2) radius �� = �

A diameter of Circle Q has endpoints (2, -3) and (6, -3).Which point also lies on circle P?A. (0, -2)B. (4, -1)C. (5,4) D. (-1, 5)

First, find center of circle, midpoint of diameter.���

�, ���(��)

�= �

�, ��)

�=(�, − �)

Then, find radius, half the distance between the endpoints of diameterdiameter 6-2= 4 radius =2

Now write the equation of the circle (� − �)�+ (� − �)�= �� (� − �)�+ (� + �)�= �Finally, try each ordered pair in the equation of the circle (� − �)�+ (� + �)�= �until you find a true statementTry (0, -2) ��(� − �)�+ (− � + �)� = � ? 17 ≠ �NOTry (4, −1) ��(� − �)�+ (− � + �)� = 4 ? 4 = 4 YESTherefore point (4, -1) also lies on circle Q.

Homework 1 : What is the center and radius of the circle given by the equation:

Homework 2: Write the equation of the circle in standard form

(� − �)�+ (� − �)�= ��

Homework 3- Constructions:Do each construction with compass and straightedge only. Describe each step with accurate geometric terms v Circle inscribed in trianglev Square inscribed in circlev Equilateral triangle inscribed in circlev Regular hexagon inscribed in circle