A tangent and a secant are drawn to circle O from an external point P. The tangent intersects the circle at Q, and the secant at R and S.
If mQR:mRS:mSQ = 2:3:4, find:
a. mQR
Q
b. mRS
c. mSQ
d. m<P
O
e. m<PQR
S
f. m<PRQ
P
R
Ex2: Two tangents are drawn to a circle from an external point R such that m<R = 70. Find the degree measures of the major arc and the minor arc into which the circle is divided by the points of tangency.
11­6 Measures of chords, tangent segments, and secant segments
*Corresponding sides of similar triangles are in proportion.
*Two triangles are similar if two angles of one triangle are congruent to two angles of the other.
(AA proves similarity)
Thm 68: If two chords intersect within a circle, the product of the measures of the segments of one chord equals the product of the measures of the segments of the other.
A
(AE)(EB)=(DE)(EC)
D
E
C
B
** use similar triangles to prove this relationship.
Ex2: Chords AB and CD intersect in a circle at point E. If AE=6, EB=10, and ED=12, find CE.
A
C
E
B
D
Thm 69: If a tangent and a secant are drawn to a circle from an external point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external segment.
Thm 70: If two secants intersect outside a circle, then the product of the measures of one secant segment and its external segment is equal to the product of the other secant segment and its external segment.
*Think of both theorems this way: (whole)(external part)=(whole)(external part)
Tangent and Secant
A
P
Two secants
A
B
B
P
C
C
D
(PA)(PA) = (PC)(PB)
(PA)(PB)=(PD)(PC)
OR
(PA)2 = (PC)(PB)
** When you have a tangent...the whole segment is the external part of the segment!
Ex1: In the accompanying diagram, PC is tangent to the circle at C, and PAB is a secant with PA = 2 and AB = 3
C
P
A
B
a. Find the exact length of PC. b. Find PC to the nearest hundreth
Section 7: Circles in the Coordinate Plane
(x­h)2+(y­k)2=r2
Ex: If the equation of a circle is (x­1)2+(y­5)2 = 36 what are two points on the circle?
Ex: The equation of a circle is x2+y2=50. What is the length of radius?
section 7 pg586­587 #32 (concurrence of perp. bisectors), 33(concurrence of angle bisectors),
Section 8: Tangents and Secants in the Coord Plane
Finding the intersection of 2 lines .... SUBSTITUTE
Ex: The line x+y=2 is a secant to circle x2+y2=100. The line y=10 is a tangent to the circle. What are the coordinates of the point of intersection of the secant and the tangent?
HW: section 6 pg 580­581
section 7 pg586­587 #29,30 and #34?
section 8 pg592­593 #3,4,15­18,20abc,23,