1. Circles Unit 6: Lesson 1 Lines that Intersect on a Circle Holt Geometry Texas ©2007

2. Objectives and Student Expectations
TEKS: G2B, G8C, G9C
The student will make conjectures about lines, angles, and circles, choosing a variety of approaches such as coordinate, transformational, and axiomatic.
The student will be expected to use Pythagorean Theorem.
The student will test conjectures about the properties and attributes of circles and the lines that intersect.

3. The interior of a circle is the set of all points inside the circle
The interior of a circle is the set of all points inside the circle. The exterior of a circle is the set of all points outside the circle.
Interior
Exterior

5. tangent
A _________________ line touches the circle at exactly one point.
A _________________ line touches the circle at exactly two points.
secant

6. Example: 1 Identify each line or segment that intersects E.
Chord: __________
Radius:__________
Secant:__________
Diameter: ________
Tangent:_________
Point of Tangency: ____

7. Example: 2 Identify each line or segment that intersects L.
Chord: __________
Radius:__________
Secant:__________
Diameter: ________
Tangent:_________
Point of Tangency: ____

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

10. 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:____
Radius of S: ____
Point of Tangency: __________
Equation for Tangent line: ____________

12. Example: 4
The Apollo 11 spacecraft orbited Earth at an altitude of 120 miles. What was the distance from the spacecraft to Earth’s horizon? Round to the nearest mile.

13. Example: 4 continued EC = CD + ED Seg. Add. Post.
Substitute 4000 for CD and 120 for ED.
EC = = 4120 mi
(EC)2 = (EH)² + (CH)2
Pyth. Thm.
41202 = (EH)
Substitute the given values.
974,400 = (EH)2
Subtract from both sides.
987 mi  EH
Take the square root of both sides.

14. Understand the Problem
Example: 5
Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall. What is the distance from the summit of Kilimanjaro to the horizon to the nearest mile? 1
Understand the Problem
The answer will be the length of an imaginary segment from the summit of Kilimanjaro to the Earth’s horizon.

15. Example: 5 cont. 2 Make a Plan
Draw a sketch. 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.

16. Example: 5 cont. Solve 3 ED = 19,340 Given Change ft to mi.
EC = CD + ED
Seg. Add. Post. =
= mi
Substitute 4000 for CD and 3.66 for ED.
(EC)2 = (EH)2 + (CH)2
Pyth. Thm.
= (EH)
Substitute the given values.
29,293 = (EH)2
Subtract from both sides.
Take the square root of both sides.
171  EH

17. Example: 5 cont. Look Back 4
The problem asks for the distance from the summit of Kilimanjaro to the horizon to the nearest mile. Check if your answer is reasonable by using the Pythagorean Theorem. Is  40042?
Yes, 16,029,241  16,032,016.
Note: these are not equal because you rounded at
the beginning when converting feet to miles.

19. Example: 6 HK and HG are tangent to F. Find HG and HK.
2 segments tangent to  from same ext. point  segments .
HK = HG
Substitute 5a – 32 for HK and 4 + 2a for HG.
5a – 32 = 4 + 2a
3a – 32 = 4
Subtract 2a from both sides.
3a = 36
Add 32 to both sides.
a = 12
Divide both sides by 3.
HG = 4 + 2(12) = 28
Substitute 12 for a and simplify.
HK = 5(12) – 32 = 28
Substitute 12 for a and simplify.