10.6 & 10.7 NOTES

10.6 FINDING SEGMENT LENGTHS IN CIRCLES Segments of the chord- when 2 chords

intersect in the interior of a circle, each chord is divided into 2 segments

segments

SEGMENTS OF CHORDS THEOREM If 2 chords intersect in the interior of a circle,

then 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.

AEEC=BEED

EC

A

D

B

EXAMPLE Find x.

xx + 2

x + 1x + 4

SEGMENTS OF SECANTS THEOREM If 2 secant segments share the same

endpoint outside the circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

EAEB=ECED

C

A

B

E

D

EXAMPLE Find x.

3

4

2

x

SEGMENTS OF SECANTS AND TANGENTS THEOREMS If a secant segment and a tangent segment

share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the lengths of the tangent segment.

GJGH=GK2

HJ

G

K

EXAMPLE Solve for x.

x8

16

10.7 WRITING AND GRAPHING EQUATIONS OF CIRCLES (x,y) represents a point on a circle By the Pythagorean Theorem, x2 + y2 = r2

when the center is at the origin.6

4

2

-2

-4

-6

-10 -5 5 10

y

x

r

STANDARD EQUATION OF A CIRCLE Center of circle is at (h,k) r2 = (x - h)2 + (y - k)2

6

4

2

-2

-4

-6

-10 -5 5 10(x,y)r

(h,k)

EXAMPLE Write the equation of the circle.

6

4

2

-2

-4

-6

-10 -5 5 10

EXAMPLE Write the standard equation of a circle with

center (-2,3) and radius 3.8.

EXAMPLE Point (8,-1) is on the circle with center (4,2).

Write the standard equation of the circle.

EXAMPLE The equation of a circle is (x + 1)2 + (y – 3)2

= 4. Graph the circle.6

4

2

-2

-4

-6

-10 -5 5 10

EXAMPLE Three forest ranger stations are located at

A(-3,2), B(2,2) and C(-1,-1.5). A fire breaks out 2 miles from A, 3 miles from B and 3.5 miles from C. Find the location of the fire.

6

4

2

-2

-4

-6

-10 -5 5 10