10.1 CIRCLES and Properties of...
Transcript of 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
= _______ Major Arc Minor Arc Semi-Circle
= _______ Major Arc Minor Arc Semi-Circle
= _______ Major Arc Minor Arc Semi-Circle
= _______ 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?
Ask yourself, where is the VERTEX located?
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)
Where is the VERTEX located? _________________
EXAMPLE 1: Find the value of x. EXAMPLE 2: Find the value of x.
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
Where is the VERTEX located? _________________
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?
EXAMPLE 5: Find the value of x. EXAMPLE 6: Find the value of x.
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 1: Find the value of x. EXAMPLE 2: Find the value of x.
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
EXAMPLE 1: Find the value of x. EXAMPLE 2: Find the value of x.
EXAMPLE 3: Find the value of x. EXAMPLE 4: Find the value of x.
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: ____________
AREA: _____________ CIRCUMFERENCE: ____________
3. 25322 yx
CENTER: _____________ RADIUS: ____________
AREA: _____________ CIRCUMFERENCE: ____________
4. 6422 yx
CENTER: _____________ RADIUS: ____________
AREA: _____________ CIRCUMFERENCE: ____________
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
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!
7. 09822 yyx
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!
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:
0 1 2 intersections.
There are:
0 1 2 intersections.
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
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.
3. 1003
3
22
yx
x
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.