CC Geometry H Aim #3: What is the relationship between tangents to a circle from an outside point? Do Now: Find the value of each variable. q0 0 0 1. 2. 0 0 95 3. 4. p a c 250 0 x0 72 b0 b0 580 0 c0 y a0

THEOREM: ______

______B

A Given: Circle O, center O O Prove: Theorem, AB ≅ AC C

1. Circle O, center O 1. Given 2. Draw OB, OC, OA. 2. Two points determine a segment.

1. PS and PT are tangent segments. a) Solve for x, if SP = x2 - 5x, TP = 3x2 + 4x - 5. b) Find SP.

S P

T 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 CIRCLE

(inscribed polygon) (circumscribed polygon)

Each side of the inscribed polygon is a Each side of the circumscribed polygon ______of the circle. 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.

F G

O

C #4-6 Each polygon circumscribes the circle. Find the perimeter of the polygon.

4. 9 cm 5. 13 in.

O 10 in. O 14 in.

13 cm 16 cm

8 in. 5 m 6.

11 m 7.5 m O

5 m

7 m

7. The square is circumscribed about circle A. What is the area of the square?

8 in.

A 300

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 Internally tangent circles are tangent circles which lie on opposite sides circles which lie on the same side of of the common tangent. the common tangent. 8. Given: Externally tangent circles O and O', tangents PD, PA, and PE Prove: PD = PE

Statements Reasons D P

E A O O'

9. Find x and y.

A

y 8 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 ______CC Geometry H Date______HW #3 1. Find x. c) a) b) x2 3x­5 O 4x­40 O 2x O x ­ 18 3x

2. Find the perimeter of the circumscribed polygon. 11 ft a) b) 6 in. 9 in. 6 ft

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? 5. Find the value of each variable.

a) b) 1000 c)

b0 a0 a0 0 0 0 b 0 520 c 60 w0 y 440 d0 540 x0 0 84 c0 840

e) d) z

O O 600 y0 x0

0 130 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