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. BC ____________________
2. EA ____________________
3. DE ____________________
4. AB ____________________
5. B
____________________
6. 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.
Radius to a Tangent Relationship:
9.
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 mO=110°.
10.2 Arc Measures and Central Angles
VOCABULARY
The Measure
of Central
angles
Adjacent
Arcs
TYPES OF ARCS
MINOR ARC


ADB
m AB  ______
m
 ______
MAJOR ARC
Major Arc
Minor Arc
Semi-Circle
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
SEMI-CIRCLE
NOW, try this one!
Find the following measurements and label each arc as minor, major, or semicircle.
Given: F.
mDFC  _____

m DE  ______
Major Arc
m
Minor Arc

ABC
Major Arc
Semi-Circle
 ______
Minor Arc
Semi-Circle
Congruent Circles:
mAFB  _____
mAFD  _____


m EC  ______
Major Arc

Minor Arc
Semi-Circle
m ADB  ______
Major Arc
Minor Arc
m DAB  ______
Major Arc
Minor Arc
Semi-Circle
Congruent Arcs:
P
Semi-Circle
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.
E

Given: m AB = 120°



m AEB  _____
m AC  _____
mAOC  _____
m CBE  ______
EC  _____
OA  ______
OD  _____
AD  ______
and OC = 12
DC  _____
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
ON THE CENTER
ON THE CIRCLE
INSIDE THE CIRCLE
OUTSIDE THE CIRCLE
OUTSIDE THE CIRCLE (T-T)
EXTRA PRACTICE:
TYPE OF ANGLE
RULE
10.6 Find Segment Lengths in Circles
Segments of Chords
EXAMPLE 1: Find the value of x.
EXAMPLE 3: Given: AB = 8, DE = 3, and EC = 4. Find the length of AE.
EXAMPLE 2: Find the value of x.
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.
x  12   y  42  16
CENTER: _____________
AREA: _____________
3.
RADIUS: ____________
CIRCUMFERENCE: ____________
x 2   y  3  25
2
CENTER: _____________
AREA: _____________
2.
CENTER: _____________
AREA: _____________
4.
RADIUS: ____________
CIRCUMFERENCE: ____________
x  22   y  52  9
RADIUS: ____________
CIRCUMFERENCE: ____________
x 2  y 2  64
CENTER: _____________
AREA: _____________
RADIUS: ____________
CIRCUMFERENCE: ____________
Write the equation of each circle using the given information.
5. Center: (-2, 3) and the radius is 3
6. Center: (-3, 5) and the radius is 10
EQUATION: ____________________________________________
EQUATION: ____________________________________________
7.
8. The endpoints of a diameter are (-2, 1) and (8, 25).
CENTER:
RADIUS:
CENTER: _____________
RADIUS: ____________
CENTER: _____________
EQUATION: ____________________________________________
RADIUS: ____________
EQUATION: ____________________________________________
What if a circle equation isn’t in standard form?
The circle equation is made up of 2 perfect square trinomials (PST), which is why it looks like the
following:
x  22   y  52  9
A PST is a trinomial that when you factor it, the factors are identical so you can write it in the following
form: (
) 2.
Complete the
Square
x  22  x  2x  2
 y  52   y  5 y  5
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.
x 2  6x  9
2.
x 2  14 x  49
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.
x 2  16 x  c
c: ________
4.
Trinomial: ______________________________
Factored Form: _______________________
x 2  3x  c
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.
2
2
STEPS:
5. x  14 x  y  2 y  40
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!
EQUATION: ______________________________________________
CENTER: _____________
RADIUS: _________________
6.
x2  y 2  10 x  8 y  16  0
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!
EQUATION: ______________________________________________
CENTER: _____________
7.
RADIUS: _________________
STEPS:
1) Move the constant (the number without a
variable) to the side of the equation
opposite the terms with variables.
x2  y2  8y  9  0
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!
EQUATION: ______________________________________________
CENTER: _____________
RADIUS: _________________
Below are the possibilities for the intersection of a line and a circle:
SYSTEMS
INVOLVING
CIRCLES AND
LINES
The line is called a ______________
The line is called a ______________
so there are:
so there are:
0
1
2 intersections.
0
1
There are:
2 intersections.
0
1
2 intersections.
Use Substitution to solve the following systems.
1.
y  x 1
x 2  y 2  25
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:
x2  9
x  3
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.
y  x2
x 2  y 2  100
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:
x2  9
x  3
3.
x3
x  32  y 2  100
4)
Plug your answers from STEP 3 into the
LINEAR EQUATION to find the second
variable.
5)
Write your answer(s) as ordered pairs.
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:
x2  9
x  3
4)
Plug your answers from STEP 3 into the
LINEAR EQUATION to find the second
variable.
5)
Write your answer(s) as ordered pairs.