10.1 circles and properties of...

10.1 CIRCLES and Properties of Tangents

CIRCLE

A circle with center P is called “circle P” is written _________.

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.

1. ____________________

2. EA ____________________

3. DE ____________________ 4. ____________________

5. B ____________________

6. ____________________

BC

AB

FA

Possibilities for Coplanar Circles:

2 POINTS OF INTERSECTION 1 POINT OF INTERSECTION NO POINTS OF INTERSECTION

Common Tangents:

Line of Centers:

COMMON EXTERNAL TANGENT

COMMON INTERNAL TANGENT

Tell how many common tangents the circles have and draw them.

7. 8. 9.

Radius to a Tangent Relationship:

PRACTICE:

10. PT is tangent to O. Find PT.

11. In the diagram, B is a point of tangency. Find the length of the radius of C.

2 Tangent to a Circle Relationship: PRACTICE:

12. RS is tangent to C at S and RT is tangent to C at T. Find the value(s) of x.

13. PS and PT are tangent to O. Find the measures of P, S, and T if the mO=110°.

10.2 Arc Measures and Central Angles

VOCABULARY

The Measure

of Central

angles

Adjacent

Arcs

TYPES OF ARCS

MINOR ARC MAJOR ARC SEMI-CIRCLE

______ABm

Major Arc Minor Arc Semi-Circle

______ADBm

Major Arc Minor Arc Semi-Circle

Given P. Find each measure.

= _______ Major Arc Minor Arc Semi-Circle





NOW, try this one! Find the following measurements and label each arc as minor, major, or semicircle. Given: F.

Congruent Circles: Congruent Arcs:

_______________ AFDmAFBmDFCm

______ABCm

______DABm

Major Arc Minor Arc Semi-Circle Major Arc Minor Arc Semi-Circle

______DEm

______ECm

______ADBm

Major Arc Minor Arc Semi-Circle Major Arc Minor Arc Semi-Circle Major Arc Minor Arc Semi-Circle

P

10.3 Apply Properties of Chords

ANY CHORD DIVIDES A CIRCLE INTO TWO ARCS

Chord Relationships:

Congruent Chords «-----» Congruent Arcs

EXAMPLE 1: Find the value of x. EXAMPLE 2: Find the value of x.

EXAMPLE 3: Find EXAMPLE 4: Find

Radius to a Chord «-----» Perpendicular Bisector (Arc too!)

EXAMPLE 1: Find QS. EXAMPLE 2: Find the measure of arc PSR.

EXAMPLE 3: Find the value of x. EXAMPLE 4: Suppose a chord of a circle is 8 inches from the center

and is 30 inches long. Find the length of the radius. DRAW A PICTURE!

EXAMPLE 5: COMBO! Find the following in O.

_____

___________

___________

___________

__________

DC

ADOD

OAEC

CBEmAOCm

ACmAEBm

Given: ABm

= 120°

and OC = 12

E

Congruent Chords «-----» Equidistant from Center

EXAMPLE 1: Find DC in M. EXAMPLE 2: Find the value of x.

10.4 Use Inscribed Angles and Polygons

Inscribed Angles «-----» Intercepted Arcs

VERTEX IS __________________________

EXAMPLE 1: Find the measure of <A. EXAMPLE 2: Find the measure of arc BC.

EXAMPLE 3: Find the measure of <B. EXAMPLE 4: Find the measure of <C.

If a right triangle is inscribed in a circle, then the hypotenuse is a DIAMTER of the circle!

EXAMPLE 5: Find the measure of <HGJ. EXAMPLE 6: Name two pairs of congruent angles.

If two inscribed angles of a circle intercept the same arc, then the angles are congruent!

Inscribed «-----» Circumscribed

Inscribed Quadrilateral «-----» Opposite Angles Supplementary

EXAMPLE 1: Find the value of x and y. EXAMPLE 2: Could a circle be circumscribed about the quadrilateral?

The only way a quadrilateral can be inscribed in a circle is if the opposite angles are supplementary!

10.5 Apply other Angle Relationships in Circles

INTERSECTING LINES AND CIRCLES: If two lines intersect in a circle, they form 3 different types of angles as shown below. NOTICE WHERE EACH VERTEX IS . . .

Ask yourself, where is the VERTEX located?



TANGENT-CHORD ANGLES

Where is the VERTEX located? _________________

EXAMPLE 1: Given line m is tangent. Find the measure of <1 EXAMPLE 2: Given the line is tangent. Find the value of x.

Make sure the angle is a combination of a tangent and a chord, NOT A SECANT AND A CHORD!

CHORD-CHORD ANGLES (The BOW-TIE ANGLE)



Notice in Example 2, chords are just parts of secants so it is still a CHORD-CHORD angle!

EXAMPLE 3: Find the measure of <1. EXAMPLE 4: Find the value of x.

EXAMPLE 5: COMBINATION PROBLEM: Given line m is tangent. Find the measures of <1, <2, <3, and <4.

<1: where is the vertex? __________ so the rule is _______________; m<1 = _______ <2: where is the vertex? __________ so the rule is _______________; m<2 = _______ <3: where is the vertex? __________ so the rule is _______________; m<3 = _______ <4: where is the vertex? __________ so the rule is _______________; m<4 = _______

TANGENT-SECANT ANGLES SECANT-SECANT ANGLES

TANGENT-TANGENT ANGLES


EXAMPLE 1: Find the value of x. EXAMPLE 2: Find the value of y.

EXAMPLE 3: Find the measure of arcs 1 and 2. EXAMPLE 4: Find the measure of <1.

What pattern do you notice with the TANGENT-TANGENT angles from Example 3 and 4?


With all of these types of angles, you can see that it is really important that you can distinguish between the types so keep asking yourself:

WHERE IS THE VERTEX?

VERTEX LOCATION TYPE OF ANGLE RULE

ON THE CENTER

ON THE CIRCLE

INSIDE THE CIRCLE

OUTSIDE THE CIRCLE

OUTSIDE THE CIRCLE (T-T)

EXTRA PRACTICE:

10.6 Find Segment Lengths in Circles

Segments of Chords


EXAMPLE 3: Given: AB = 8, DE = 3, and EC = 4. Find the length of AE.

Segments of Secants & Segments of Secants/Tangents

Segments of Secants

Segments of Secants & Tangents



10.7 Write and Graph Equations of Circles

Standard

Equation of a

Circle

The standard equation of a circle with center (h, k) and radius r is:

Given the equation for each circle, determine the center, radius, and graph the circle. Find the EXACT area and the circumference of each.

1. 164122 yx

CENTER: _____________ RADIUS: ____________

AREA: _____________ CIRCUMFERENCE: ____________

2. 95222 yx

CENTER: _____________ RADIUS: ____________


3. 25322 yx

CENTER: _____________ RADIUS: ____________


4. 6422 yx

CENTER: _____________ RADIUS: ____________


Write the equation of each circle using the given information.

5. Center: (-2, 3) and the radius is 3

EQUATION: ____________________________________________

6. Center: (-3, 5) and the radius is 10

EQUATION: ____________________________________________

7.

CENTER: _____________ RADIUS: ____________ EQUATION: ____________________________________________

8. The endpoints of a diameter are (-2, 1) and (8, 25). CENTER: RADIUS: CENTER: _____________ RADIUS: ____________ EQUATION: ____________________________________________

What if a circle equation isn’t in standard form?

Complete the

Square

The circle equation is made up of 2 perfect square trinomials (PST), which is why it looks like the following:

95222 yx

A PST is a trinomial that when you factor it, the factors are identical so you can write it in the following form: ( )2.

555

222

2

2

yyy

xxx

If a circle equation isn’t in standard form, we must CREATE 2 perfect square trinomials to put it into standard form. We do this by completing the square.

Let’s first review factoring a PST.

1. 962 xx

2. 49142 xx

Now let’s MAKE a PST by using the complete the square method. Find the value of c that will make each a PST. Then factor the trinomial. To find c: Given any trinomial, as long as we have a b value, we can create a perfect square trinomial. If we take our b value, divide it by 2, then square it, we will always get our c value.

3. cxx 162

c: ________ Trinomial: ______________________________ Factored Form: _______________________

4. cxx 32

c: ________ Trinomial: ______________________________ Factored Form: _______________________

Now let’s change a circle equation into Standard form using the complete the square method. Don’t forget from Algebra I: if you add a number to one side of an equation, you must add it to the other side to balance the equation.

5. 40214 22 yyxx

EQUATION: ______________________________________________ CENTER: _____________ RADIUS: _________________

STEPS: 1) Move the constant (the number without a

variable) to the side of the equation opposite the terms with variables.

2) Separate the x terms from the y terms.

3) Make a PST with the x terms by completing the square and then do the same for the y terms. The only time you need to complete the square is if there is an x2 term and an x term. If there is only an x2 term, then you can leave it alone. BALANCE YOUR EQUATION!

4) Factor both PST so they are in the form: ( )2

5) Combine all of the constants on the opposite side of the equation.

6) Now that your equation is in standard form, you can easily find the center and the radius!

6. 01681022 yxyx









7. 09822 yyx









SYSTEMS

INVOLVING

CIRCLES AND

LINES

Below are the possibilities for the intersection of a line and a circle:

The line is called a ______________ so there are:

0 1 2 intersections.

The line is called a ______________ so there are:


There are:


Use Substitution to solve the following systems.

1. 25

1

22

yx

xy

STEPS: 1) Solve the LINEAR EQUATION for either

the x or y variable.

2) Substitute the linear equation into the variable you solved for from STEP 1 in the CIRCLE EQUATION.

3) Solve the circle equation for the variable. Don’t forget that when you get and equation like the following, you get two answers:

3

92

x

x

4) Plug your answers from STEP 3 into the

LINEAR EQUATION to find the second variable.

5) Write your answer(s) as ordered pairs.

2. 100

2

22

yx

xy





3

92

x

x




3. 1003

3

22

yx

x





3

92

x

x




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