Warm-up

In the diagram, chords AB and CD are parallel.

Prove that AC is congruent to BD.

Theorem: In a circle, parallel chords intercept congruent arcs.

Theorem: An angle formed by two chords intersecting inside a circle is equal to half the sum of the intercepted arcs.

ARE = ½ (AE + DC)

Theorem: An angle formed by two secants, two tangents, or by a secant and

a tangent drawn from a point outside a circle is equal to half the difference

of the intercepted arcs.

Theorem: The measure of an angle formed by a tangent and a chord of a circle is half the measure of the arc between them.

ABC = ½ AB

APE = ½ (AE – AC)APE = ½ (AE – CB)

P

z

y

40x

x = 140

y = 70

z = 40

A

C B

Triangle ABC is an isosceles triangle with AB AC. Find the measures of x, y, and z.

65

2

9

10

14

3

8

7

D

P

H

F

C

E

B

A

is tangent to circle P; is a diameter;

AB = 30, CD = 40, DE =50

Find the measure of each numbered angle.

EF AD

30

40

50

65

2

9

10

14

3

8

7

D

P

H

F

C

E

B

A

is tangent to circle P; is a diameter;

AB = 30, CD = 40, DE =50

Find the measure of each numbered angle.

EF AD

1 = 40

30

40

50

110

130

2 = 15

3 = 25

4 = 40

5 = 65

6 = 100

7 = 45

8 = 30

9 = 25

10 = 65

Part II

In the diagram, WS is tangent to P. 1. Name two radii. 2. Name a diameter. 3. Name three chords. 4. Name an inscribed angle. 5. Name two right angles. 6. Name an angle congruent to R.

.

W

PRS

T

In the diagram, EF , CF , and P are tangent to A, and EF = 16 cm. 7. PB = _______ cm 8. AC = ______ cm 9. mACF = _______ 10. FC = ______ cm

. .

A

B

F

P

E

C

8 4

90 8

In P, mWPY = 91 , and mWX = 48 . 11. mXY = _______ 12. mWZY = _______ 13. mZPY = ________ 14. mWZ = _______

.

P

Z

XW

Y

43 269

137 132

In the diagram, MN is tangent to the circle. 15. mLMK = _______ 16. mLMN = _______ 17. JMK _______ 18. JML is supplementary to ______

38

122

L

K

M

J

N

91 61

JLK JKL

B and D are supplementary.

The median is half the length of the hypotenuse.

D

A

B C

Theorem: Opposite angles of an inscribed quadrilateral are supplementary.

Part I 1. a. Use Geometer’s Sketchpad to construct a quadrilateral inscribed in a circle, as shown. b. Make a conjecture about the relationship between the measure of B and the measure of D. c. Prove your conjecture.

D

A

B C

(Con)Cyclic Quadrilaterals

(Con)Cyclic quadrilateral is a quadrilateral that may be inscribed in a circle

The converse is also true:If the opposite angles of a quadrilateral aresupplementary, the quadrilateral is cyclic.

We just proved that the opposite angles of a cyclic quadrilateral are supplementary.

B

A

E

D

C

In the diagram, radius AB is perpendicular to radius AC, and CD isperpendicular to ray BD.

This is the diagram for question 2 in the homework

Are any four points in this diagram the vertices of a cyclic quadrilateral?

B

A

E

D

C

In the diagram, radius AB is perpendicular to radius AC, and CD isperpendicular to ray BD.

This is the diagram for question 2 in the homework

Are any four points in this diagram the vertices of a cyclic quadrilateral?

2. In the diagram, radius AB is perpendicular to radius AC .

Point E is chosen randomly on minor arc BC and CD is constructed perpendicular to ray BE at point D. Using Geometer’s Sketchpad, make a conjecture about the relationship between the lengths of segments CD and DE and prove you are correct.

A

B

E D

C

B

A

E

D

C

The median is half the length of the hypotenuse.

 3. a. Use Geometer’s Sketchpad to construct a right triangle. b. Construct the median to the hypotenuse of the right triangle.  c. Make a conjecture about the relationship between the length of the hypotenuse and the length of the median.

Theorem: The median to the hypotenuse of a right triangle is half the length of the hypotenuse.

Suppose it doesn’t.

Claim: A circle with diameter AC also passes through point B.

Theorem: The median to the hypotenuse of a right triangle is half the length of the hypotenuse.

Theorem: The median to the hypotenuse of a right triangle is half the length of the hypotenuse.

Theorem: When two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

(AP)(PB) = (CP)(PD)

A new twist on a familiar problem

Infinitely many rectangles with different dimensions have an area of 36 square units. Use Geometer’s Sketchpad to construct a rectangle whose area is 36, and which retains that area when the dimensions are changed.

Theorem: When a tangent and a secant intersect outside a circle, the square of the tangent’s length is equal to the product of the lengths of the secant and the length of its external segment. (Known as The Power of the Tangent)

(AP)2 = (CP)(PD)

12

10

x

8

Theorem: When a tangent and a secant intersect outside a circle, the square of the tangent’s length is equal to the product of the lengths of the secant and the length of its external segment. (Known as The Power of the Tangent)

(AP)2 = (CP)(PD)

In the diagram, PC is tangent to circle A, and secant segment PB crosses diameter EC at point N. If BN = 11, ND = 9, and PD = 16,

1.What is the length of PC?

2. What is the length of the radius of circle A?

Application 1

7

74 10.57

24

Theorem: When a 2 secants intersect outside a circle, the product of the length of one secant and the length of its external segment is equal to the product of the length of the other secant and the length of its external segment.

(AP)(PB) = (CP)(PD)

10

8

7

x

9

Simulate the following situation using Geometer’s Sketchpad.The families living in the 3 houses shown below chipped in to buy a swing set for their kids. They want to place the swing set so that it is the same distance from all three houses. Where should the swing set be placed?

The circumcenter of a triangle is the center of the circumscribed circle.

It is the intersection of the perpendicular bisectors of the sides of the triangle.

The circumcenter of a triangle is equidistant from the vertices of the triangle.

Simulate the following situation using Geometer’s Sketchpad.You are a lifeguard on a small island that is roughly shaped like a triangle. You need to station yourself so that you are as close to each shoreline as possible. Where should you place your chair?

The incenter of a triangle is the center of the inscribed circle.

It is the intersection of the angle bisectors of the angles of the triangle.

The incenter of a triangle is equidistant from the sides of the triangle.

The incenter of a triangle is the center of the inscribed circle.

It is the intersection of the angle bisectors of the angles of the triangle.

The incenter of a triangle is equidistant from the sides of the triangle.

The orthocenter of a triangle is the intersection of the altitudes.

The 3 altitudes of ABC

O

orthocenter O

The three perpendicular bisectors of ABC

O

P

orthocenter Ocircumcenter P

The three medians of ABC

O

PR

circumcenter Porthocenter O

The Euler Line

centroid R

O

PR

circumcenter Porthocenter O

The Euler Line

centroid R

The Nine Point Circle

Euler Line

Midpoint of segment from orthocenter to incenter (center of nine point circle

The Nine Point Circle

Circumcenter