1. Section 11-1 Lines that Intersect Circles

2. Vocabulary Interior of a Circle: All points inside the circle
Exterior of a Circle: All points outside the circle

3. Identify each line or segment that intersects L.
Example 1:
Identify each line or segment that intersects L.
chords:
secant:
tangent:
diameter:
radii:
JM and KM JM m KM
LK, LJ, and LM

4. Identify each line or segment that intersects P.
Example 2
Identify each line or segment that intersects P.
chords:
secant:
tangent:
diameter:
radii:
QR and ST ST UV ST
PQ, PT, and PS

6. Equation of tangent line: y = 0
Example 3:
Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point.
radius of R: 2
Center is (–2, –2). Point on  is (–2,0). Distance between the 2 points is 2.
Equation of tangent line: y = 0
radius of S: 1.5
Horizontal line through (–2,0)
Center is (–2, 1.5). Point on  is (–2,0). Distance between the 2 points is 1.5.
point of tangency: (–2, 0)
Point where the s and tangent line intersect

7. Equation of tangent line: y = –1
Example 4
Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point.
radius of C: 1
Center is (2, –2). Point on  is (2, –1). Distance between the 2 points is 1.
Equation of tangent line: y = –1
radius of D: 3
Horizontal line through (2,-1)
Center is (2, 2). Point on  is (2, –1). Distance between the 2 points is 3.
Pt. of tangency: (2, –1)
Point where the s and tangent line intersect

8. A common tangent is a line that is tangent to two circles.

9. Tangent Theorems…

10. Step 1: Sketch the circle
Example 5:
Early in its flight, the Apollo 11 spacecraft orbited Earth at an altitude of 120 miles. What was the distance from the spacecraft to Earth’s horizon rounded to the nearest mile?
*Hint: Earth’s Radius = 4000 miles*
Step 1: Sketch the circle
Let C be the center of Earth, E be the spacecraft, and H be a point on the horizon. You need to find the length of EH, which is tangent to C at H. Since we know that tangent lines are perpendicular to the radius of a circle, Angle H must be 90 degrees
120 mi
4000 mi
In order to solve, we’ll need to find the length of EC
EC = CD (radius of circle) + ED (Altitude)
= = 4120 mi

11. Step 2: Use Pythagorean Theorem to Solve
EC2 = EH² + CH2
Pyth. Thm.
41202 = EH
974,400 = EH2
987 mi  EH

12. Step 1: Sketch the circle
Example 6
Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall. What is the distance from the summit of Kilimanjaro to the horizon of the Earth to the nearest mile? *Hint: The Earth’s radius is 4000 miles* *Hint: There are 5,280 feet in every mile*
Step 1: Sketch the circle
Let C be the center of Earth, E be the summit of Kilimanjaro, and H be a point on the horizon. You need to find the length of EH, which is tangent to C at H. By Theorem , EH  CH.
So ∆CHE is a right triangle.
First, convert feet to miles…
ED = 19,340

13. Step 2: Use Pythagorean Theorem to Solve
EC = CD + ED =
= mi
Step 2: Use Pythagorean Theorem to Solve
EC2 = EH2 + CH2
= EH
29,293 = EH2
171 miles  EH

15. Example 7:
HK and HG are tangent to F. Find HG.
HK = HG
5a – 32 = 4 + 2a
3a – 32 = 4
3a = 36
a = 12
HG = 4 + 2(12)
= 28

16. Example 8
RS and RT are tangent to Q. Find RS.
RS = RT
x = 4x – 25.2
–3x = –25.2
x = 8.4
= 2.1

17. Assignment #53
Pg. 751
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