12.1B Lines that Intersect Circles
Objectives: G.C.2: Identify and describe relationships among inscribed angles, radii, and chords.
For the board: You will be able to identify tangents, secants, and chords and use tangents to solve
problems.
Instruction:
Theorem
If a line is tangent to a circle, then it is perpendicular
to the radius drawn to the point of tangency.
Given: line l is tangent to circle Q at P
P
l
Conclusion: line l  QP
Theorem
If a line is perpendicular to a radius of a circle at a point
on the circle, then the line is tangent to the circle.
Given: line l  QP
Q
Conclusion: line l is tangent to circle Q at P
Many things circle the earth at various heights, planes, satellites, and space craft.
Each is interested in its visible distance of sight, the distance to the horizon.
To find this you need the height at which the object is circling, and the radius of the earth.
Earth’s radius = 4000 mi.
Open the book to page 795 and read example 3.
Example: The Apollo 11 spacecraft orbited Earth at an attitude of 120 miles.
What was the distance from the spacecraft to Earth’s horizon
rounded to the nearest mile?
Hint: Earth’s radius = 4000 mi.
ED = 120 miles
CD = CH = 4000 mi
EC = 4000 + 120 = 4120 mi
EH is tangent to circle C at H (CH is a radius) therefore
CH is | HE or ΔCHE is a right triangle.
The Pythagorean Theorem applies and CD2 + EH2 = CE2.
40002 + EH2 = 41202
EH2 = 16974400 – 16000000 = 974400
EH = 987 mi
E
?
D
C
H
White Board Activity:
Practice: Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall or
about 3.66 mi tall. What is the distance from the summit of
Kilimanjaro to the horizon to the nearest mile?
KD = 19,340 ft = 3.66 mi
CD = CH = 4000 mi
KH is tangent to circle C at H (CH is a radius) therefore
KH is | CE or ΔCHK is a right triangle.
CH2 + HK2 = CK2
40002 + HK2 = (3.66 + 4000)2
HK2 = 29293.4
HK = 171 mi.
K
?
D
H
C
Theorem
If two segments are tangent to a circle from the same external point,
then the segments are congruent.
Given: SR and ST are tangent to circle P
Conclusion: SR  ST.
S
R
P
T
H
Open the book to page 796 and read example 4.
Example: HK and HG are tangent to circle F. Find HG.
5a – 32 = 4 + 2a
3a – 32 = 4
3a = 36
a = 12
HG = 4 + 2(12) = 4 + 24 = 28
4 + 2a
G
5a - 32
P
K
White Board Activity:
Practice: RS and RT are tangent to circle Q. Find RS.
a.
R
b.
R
x – 6.3
n+3
T
T
2n - 1
x/4
P
P
S
x/4 = x – 6.3
6.3 = ¾ x
x = 8.4
RS = RT = 2.1
S
2n – 1 = n + 3
n=4
RS = RT = 7
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 797 – 798 prob. 8 – 10, 15 – 17, 26, 27, 31 – 33.
Hints: 31. (1) QR and QS are radii of circle Q so they have the same measure.
32. AB & AD, AC & AE are tangents to the circles from a point outside the circle.
33. KJ & JL, JL & JM are tangents to the circles from a point outside the circle.
For a Grade:
Text: pgs. 797 – 798 prob. 8, 16, 26.