Section 11-1 Lines that Intersect Circles

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

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

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

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

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

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

Tangent Theorems…

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) = 4000 + 120 = 4120 mi

Step 2: Use Pythagorean Theorem to Solve EC2 = EH² + CH2 Pyth. Thm. 41202 = EH2 + 40002 974,400 = EH2 987 mi  EH

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 11-1-1, EH  CH. So ∆CHE is a right triangle. First, convert feet to miles… ED = 19,340

Step 2: Use Pythagorean Theorem to Solve EC = CD + ED = 4000 + 3.66 = 4003.66mi Step 2: Use Pythagorean Theorem to Solve EC2 = EH2 + CH2 4003.662 = EH2 + 40002 29,293 = EH2 171 miles  EH

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

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

Assignment #53 Pg. 751 #1-10 all