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Tangents and Circles, Part 1
Lesson 58
Definitions (Review)
A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle.
Radius – the segment (distance) from the center to a point on the circle
Congruent circles – circles that have the same radius. Diameter – the segment with endpoints on the circle
and contains the center
Diagram of Important Terms (Review)
diameter
radiusP
center
name of circle: P
Definition (Review)
Chord – a segment whose endpoints are points on the circle.
AB is a chord
B
A
Definition (Review)
Secant – a line that intersects a circle in two points.
MN is a secant
N
M
Definition (Review)
Tangent – a line in the plane of a circle that intersects the circle in exactly one point.
ST is a tangent
S
T
Definition (Review)
Point of tangency – the point at which a tangent line intersects the circle to which it is tangent
point of tangency
Review
Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.
FC
B
G
A
H
D
E
Id. CE
c. DF
b. EI
a. AH tangent at Point B
diameter
chord
radius
Theorem 58-1
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. l
Q
P
If line l is tangent to ʘQ at point P, then line l ┴
Example 1
Line r is tangent to ʘC at point B and line s passes through point C. Lines r and s intersect at point A.
a) Sketch and label.b) What is m<CBA?c) If m<BCA = 50°,
then what is m<CAB?
Example 1a
Line r is tangent to ʘC at point B and line s passes through point C. Lines r and s intersect at point A.
a) Sketch and label.
Example 1a
Line r is tangent to ʘC at point B and line s passes through point C. Lines r and s intersect at point A.
a) Sketch and label.
Example 1a
Line r is tangent to ʘC at point B and line s passes through point C. Lines r and s intersect at point A.
a) Sketch and label.
Example 1b
Line r is tangent to ʘC at point B and line s passes through point C. Lines r and s intersect at point A.
b) What is m<CBA?
m<CBA = 90°
Example 1c
Line r is tangent to ʘC at point B and line s passes through point C. Lines r and s intersect at point A.c) If m<BCA = 50°, then
what is m<CAB?
m<CAB + m<BCA = 90°
m<CAB + 50° = 90°
m<CAB = 40°
Theorem 58-2
If a line in a plane is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.
l
Q
P
If line l ┴ , then line l is tangent to ʘQ.
Example 2
Tell whether CE is tangent to D.
45
43
11
D
E
C
Use the converse of the Pythagorean Theorem to see if the triangle is right.
112 + 432 ? 452
121 + 1849 ? 2025
1970 2025
CED is not right, so CE is not tangent to D.
Theorem 58-3
If two tangent segments are drawn to a circle from the same exterior point, then they are congruent.
SP
R
T
If SR and ST are tangent to P, then SR ST.
Example 3
AB is tangent to C at B.AD is tangent to C at D.
Find the value of x.
11
x2 + 2
AC
D
BAD = AB
x2 + 2 = 11
x2 = 9
x = 3
Example 4
a) Find the perimeter of ΔJLM.
b) What is the relationship between ΔJLM and ΔJLK?
c) What is the perimeter of JKLM?
Example 4a
a) Find the perimeter of ΔJLM.
MJ = 12 in52 + 122 = (LJ)2
25 + 144 = (LJ)2
169 = (LJ)2
13 in = LJ
P = 12 + 5 +13P = 30 in
Example 4b
b) What is the relationship between ΔJLM and ΔJLK?
Therefore, ΔJLM ΔJLK by SSS
Example 4c
c) What is the perimeter of JKLM?
P = 5 + 5 + 12 + 12P = 34 in
Do you have any questions?
