12-1: Lines that Intersect Circles

12-2: Arcs and Chords

12-4: Inscribed Angles

**Analytic geometry**

**By: Brenda Ontivelos**

**Module 12**

MCC9-12.G.C5

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of the sector.

Burger, Chard, Kennedy, Leinward, Renfro, Roby, Seymour, and Bert K. Waits.

Georgia Analytic Geometry

. Orlando: Houghton Mifflin Harcourt Publishing Company, 2014. Textbook

**Resources**

**Write the primary idea of the mind map in the center. Use different color notes to differentiate between topics. Use lines and arrows to create branches that connect ideas to each other.**

Citation for Workbook

14-1: Factoring x(squared) + bx + c

14-3: Factoring Special products

14-2: factoring ax(squared) + bx + c

**Module 14**

MCC9-12.A.SSE.1, MCC9-12.A.SSE.1a, MCC9-12.A.SSE.1b, MCC9-12.A.SSE.2

Interpret expressions that represent a quantity in terms of its context.

a.Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entiety.

Use the structure of an expression to identify ways to rewrite it

17-2: Circles in the Coordinate Plane

17-1: Introduction to Coordinate Proof

17-3: Parabolas

**Module 17**

MCC9-12.G.GPE.4

Use coordinates to prove simple geometric theorems algebraically.

16-1: Solving quadratic equations by graphing and factoring

16-2: Completing the square

**Module 16**

MCC9-12.A.REI.4b

Solve quadratic equations by inspection (e.g.,for x =49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b.

10-1 Trigonometric Ratios

Essential question:

How can you use inverse trigonometric functions to solve for the angles in a right triangle?

Essential question:

How can trigonometric ratios be used to estimate distances when you know an angle of elevation or depression?

**Module 10: Trigonometry**

MCC9-12.G.SRT.6, MCC9-12.GSRT.7, MCC9-12.G.SRT.8

Understand that by similarity, side ratios in right triangles are properties of the angles in the the triangle, leading to the definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

Essential Question:

How do side ratios in right triangles define trigonometric ratios?

Vocabulary

Trigonometric Ratio:

A ratio of two sides of a right triangle

Sine:

In a right triangle, the ratio of the length of the leg opposite A to the length of the hypotenuse.

Cosine:

In a right triangle, the cotangent

10-2: Solving right Triangles

10-3: Angles of elevation and depression

Vocabulary

Angle of elevation:

The angle formed by a horizontal line and a line of sight to a point above

Angle of depression:

The angle formed by a horizontal line and a line of sight to a point below

Who uses this?

Contractors use this to measure the angles for ramps in buildings for the handicapped.

Pilots and air traffic controllers use it to calculate distances.

Who uses this?

Essential Question:

What are the various ways that lines circles can intersect?

Vocabulary

Interior of a circle:

The set of all points inside a circle.

Exterior of a circle:

The set of all points outside a circle.

Chord:

A segment whose endpoints lie on a circle.

Secant:

A line that intersects a circle at two points.

Tangent:

A line in the same plane as a circle that intersects it at exactly one point.

Point of Tangency:

The point where the tangent and a circle intersect.

Congruent Circles:

Two circles that have congruent radii.

Tangent Circles:

Two coplanar circles that intersect at exactly one point.

Common Tangent:

A line that is tangent to two circles

If a line is tangent to a circle then it is perpendicular to the radius drawn to the point of tangency.

If a line is perpendicular to the radius of a circle at a point in the circle, then the line is tangent to the circle.

If two segments are tangent to a circle from the same external point, then the segments are congruent.

Theorems

Essential Question:

How are arcs of circles measured in relation to central angles?

Vocabulary

Central Angle:

An angle whose vertex is the center of a circle

Arc:

An unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.

Minor Arc:

An arc whose points are on or in the exterior of a angle

Major Arc:

An arc whose points are on or in the exterior of a central angle

Semicircle:

An arc of a circle whose endpoints lie on a diameter.

Adjacent Arcs:

Two arcs of the same circle that intersect at exactly one point.

Congruent Arcs:

Two arcs that are in the same or congruent circles and have the same measure.

Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

m ABC = m AB + m BC

A

B

C

Theorem

In a circle or congruent circles:

1) Congruent central angles have congruent chords

2) Congruent chords have congruent arcs.

3) Congruent arcs have congruent central angles.

A

B

C

D

E

EAD = BAC

In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc.

In a circle, the perpendicular bisector of a chord is a radius (or diameter)

Essential Question:

How is proportional reasoning used to find areas of the circle sectors and lengths of arcs?

Vocabulary

Sector of a circle:

A region bounded by two radii of the circle and their intercepted arc.

Segment of a circle:

A region bounded by an arc and its chord.

Arc length:

The distance along an arc measured in linear units.

Who uses this?

Farmers use irrigation radii to calculate the area of sectors

Essential Question:

How is the measure of an angle inscribed in a circle related to the measure of its associated central angle?

Vocabulary

Inscribed angle:

An angle whose vertex is on a circle and whose sides contain chords of the circle.

Intercepted arc:

It consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them.

Subtends:

A segment or arc an angle if the endpoints of the segment or arc lie on the sides of the angle.

Theorems

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

If a quadrilateral is inscribed in a circle then its opposite angles are supplementary.

An inscribed angle subtends a semicircle if and only if the angle is a right angle

Essential Question:

How can you factor a trinomial with a leading coefficient of 1?

Vocabulary

Monomial:

A number, a variable, or a product of numbers and variables with the whole-number exponents.

Polynomial:

A monomial or a sum of monomials.

Binomial:

A polynomial that has two terms.

Trinomial:

A polynomial that has three terms.

Essential Question:

How can you factor a trinomial with a leading coefficient other than 1?

Essential Question:

How can you recognize and factor special products?

Essential Question:

What is a coordinate proof in geometry?

Vocabulary

Coordinate proof:

A style of proof that uses coordinate geometry and algebra.

Essential Question:

How can you write an equation for a circle in the coordinate plane with known center and radius?

Theorem

The equation of a circle with center (h,k) and the radius r is (x-h) + (y-k) =r

Essential Question:

How can you derive an equation for a parabola using a focus and a directix?

Vocabulary

Focus of a parabola:

A fixed point F used with the a directix to define a parabola.

Directix:

A fixed line used to define a parabola.

Essential Question:

How can you solve quadratic equations by graphing a related quadratic function or by factoring?

Pythagorean Theorem

AC*AC = AB*AB + BC*BC

= 8*8 + 6*6 = 100

So AC = the square root of 100 = 10

Finding the unknown measures

A

B

C

8

6

m<A = tan-1 (6/8) = 36.87

Since the acute angles of a right triangle are complementary m<C is approximately 90° - 36.87° = 53.13°

Theorems

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency

If a line is perpendicular to a radius of a circle at a point then the line is tangent to the circle.

If two segments are tangent to a circle form the same point then the segments are congruent

Find JK

4x - 1

2x + 9

J

K

L

4x - 1 = 2x + 9

+1 +1

4x = 2x + 10

-2x -2x

2x/2 = 10/2

x = 5

4x -1

4(5) - 1

20 - 1 = 19

JK = 19

CB is congruent to DE

Arc CB is congruent to Arc DE

Corollary

Find measurement of Arc EG and Angle DEF

29°

78°

F

E

G

D

Arc EG = Angle EFG * 2

= 29*2

Arc EG = 58°

Arc DF = Angle DEF *2

78°/2 = Angle DEF * 2/2

39° = Angle DEF

x(squared) + 3x - 10

Factor x(squared) + 3x - 10

1 * 10 10 - 1 =9

2 * 5 5 - 2 = 3

( x + 5 )( x - 2 )

Factor 3x (squared) - 20x + 12

3x(squared) -20x + 12

36

Factors of 36

1 * 36 36 + 1 = 37

2 * 18 18 + 2 = 20

3 * 12 12 + 3 = 15

4 * 9 9 + 4 = 13

6 * 6 6 + 6 = 12

3x(squared) - 18x - 2x + 12

3x(x - 6) -2(x - 6)

(x - 6) (3x - 2)

Vocabulary

Zero of a function:

A value of the input

x

that makes the output

f(x)

equal zero.

Root of an equation:

The values of the variable that make the equation true.

Essential Question:

How can you solve quadratic equations by using the square roots or by completing the square?

Vocabulary

Completing the square:

If a quadratic expression of the form x(squared) = bx cannot model a square, you can add a term to form a perfect trinomial.

x(squared) - 2x +

(

-2

/2)(squared) = (-1)(squared) = 1

x(squared) - 2x + 1

(x - 1)(squared)

Complete the square and write solution as a binomial squared

Find (b/2)(squared)

16-3: The quadratic Formula

Essential Question:

How can you derive the Quadratic formula and use it to solve any quadratic function?

Vocabulary

Discriminant:

A part of the quadratic formula that you can use to determine the number of real roots of a quadratic equation.

Citation for Textbook

Oppenzato, Colleen O'Donenell.

Common Core Coach Analytic Geometry

. New York: Triumph Learning, 2010. Workbook