CC Geometry 10H
Aim 3: What is the relationship between tangents to a circle from an outside point?
Do Now: Find the value of each variable.
2. 0 c0
1.
95
3.
0
a
25
4.
0
q0
p0
720
x0
b
0
b0
580
c0
y0
a0
THEOREM: ____________________________________________________________
____________________________________________________________________
B
Given: Circle O, center O
A
O
Prove:
C
1. Circle O, center O
2. Draw OB, OC, OA.
Exercises
1. Given
2.
1. PS and PT are tangent segments.
a) Solve for x, if
S
SP = x2 - 5x, TP = 3x2 + 4x - 5.
P
T
b) Find SP.
Definition:
A circle circumscribed about a polygon is a circle that passes
through each vertex of the polygon.
Definition:
A circle inscribed in a polygon is a circle that has a point of
tangency with each side of the polygon.
CIRCUMSCRIBED CIRCLE
(inscribed polygon)
INSCRIBED CIRCLE
(circumscribed polygon)
Each side of the inscribed polygon is a
_________________ of the circle.
Each side of the circumscribed polygon
is a _________________to the circle.
2. Circle O is inscribed in ΔABC. Find the perimeter of ΔABC.
B
E
8
F
A
O
10
D
15
C
3. ΔCDE is circumscribed about circle O. Find DE.
G
F
O
C
Practice
#4-6 Each polygon circumscribes the circle. Find the perimeter of the polygon.
5.
9 c
4.
13 in.
m
O
10 in.
O
14 in.
13 cm
16 cm
8 in
.
5 m
6.
11 m
7.5 m
O
m
5 7 m
7. The square is circumscribed about circle A. What is the area of the square?
8 in.
300
A
Tangent Circles
A common tangent to two circles is a line which is tangent to each of the circles.
Tangent circles are circles in a plane that are tangent to the same line at the same
point.
Externally tangent circles are tangent
circles which lie on opposite sides
of the common tangent.
Internally tangent circles are tangent
circles which lie on the same side of
the common tangent.
8. Given: Externally tangent circles O and O', common tangents PD, PA, and PE
Prove: PD = PE
Statements
P
D
Reasons
E
A
O
O'
9. Find x and y.
A
8
y
x
B
E
C
D
R
•
•
•
•
S
T
Let's Sum it Up!
A tangent to a circle intersects the circle at one and only one point.
Two tangents drawn to a circle from an external point to the points of
intersection are equal in length.
Internally tangent circles lie on the same side of the common tangent.
Externally tangent circles lie on opposite sides of the common tangent.
Name __________________
Date_________
Hwk. #3
CC Geometry H
a)
3x
­5
4x
­4
0
1. Find x.
O
b)
c)
x2 O 2x
O
x ­ 18
3x
2. Find the perimeter of the circumscribed polygon.
11 ft
9 in.
6 in.
6 ft
b)
a)
12 ft
10 ft
8 in.
5 ft
3. The three segments are tangent to the circle at points B, F, and G.
If y =
, find x, y, and z.
4. A circular pond is fenced on two opposite sides (CD,FE) with wood and the
other two sides with metal fencing. If all four sides of fencing are tangent
to the pond, is there more wood or metal fencing used?
OVER
5. Find the value of each variable.
b)
a)
b0
a0
c0
520
c)
1000
a0
b0
600
y0
w0
44
0
d0
540
x0
840
840
c0
e)
d)
z
O
O
y0
x
0
1300
600
radius = 8√3
A
6. If AO = 15, find AC and BC, to the nearest tenth.
540
O
C
B
7. Draw all common tangents and state the number of common tangents.
a)
b)
c)
d)
Review
8. Which conditions allow you to conclude that a quadrilateral is a rhombus?
(a) one pair of sides congruent and parallel, and diagonals are equal
(b) quadrilateral is equiangular
(c) diagonals bisect each other and two adjacent sides are equal
(d) both pairs of opposite sides are equal, and one angle is a right angle.
9. What is the surface area of a sphere with radius 7 cm?
(a) 196π cm2
(b)
π cm2
(c) 49π cm2
(d) 14π cm2
10. Which line or lines are perpendicular to the line y = 4x - 1?
I. y = 4x + 7
II. y = ¼x + 3
III. y = -¼x - 5
IV. x + 4y = 16
(a) I only
(b) II only
(c) III only
(d) III and IV