Lesson 9 5 Theorems

Lesson 9-5: Properties of Tangents and Secants : 

Lesson 9-5: Properties of Tangents and Secants

Theorems 9-8 : 

Theorems 9-8 If a line is tangent to a circle, then it is perpendicular to the radius drawn at the point of tangency. B C l Symbols If l is tangent to circle C at B, then l ?CB

Theorems 9-9 : 

Theorems 9-9 In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. l C B Symbols: If l ? CB, then l is tangent to circle C at B.

Example 1 Use of Properties of Tangents : 

Example 1 Use of Properties of Tangents B C 12 A 13 AC is tangent to circle B at point C. Find BC

Slide 5: 

Solution: BC is a radius of circle B, so you can apply Theorem 9-8 to conclude that BC and AC are perpendicular. So ?BCA is a right angle, and BCA is a right triangle. To find BC, use the Pythagorean Theorem.

Slide 6: 

Solution continued: (BA)² = (BC)² + (AC)² Pythagorean Theorem 13² = (BC)² + 12² Substitute 13 for BA and 12 for AC. 169 + (BC)² + 144 Multiply 25 = (BC)² Subtract 144 from each side. 5 = BC Find the positive square root.

Real World Application : 

Real World Application Silos are used as storage bins for feed for farm animals. Round silos allow for the feed to be tightly packed, which prevents it from spoiling. How are tangents related to a silo?

Slide 8: 

A r 8 ft. r B 16 ft. C You are standing at C, 8feet from a silo. The distance to a point of tangency is 16 feet. What is the radius of the silo? Solution: Tangent BC is perpendicular to radius A.B at B, so ABC is a right triangle. So, You can use the Pythagorean Theorem

Slide 9: 

(AC)² = (AB)² + (BC)² Pythagorean Theorem (r+ 8)² = r² + 16² Substitute r² + 16r + 64 = r² +256 (r+8) (r+8) = r² +16r + 64 16r +64 = 256 Subtract r² from each side. 16r = 192 Subtract 64 from each side. r= 12 Divide each side by 16. The radius of the silo is 12 feet. Solution-continued:

Slide 10: 

How can you show that EF must be tangent to D? Example 2 Verify a Tangent to a Circle Solution: Use the Converse of the Pythagorean Theorem. To determine whether DEF (DF)² ? (DE)² + (EF)² Compare (DF)² with (DE)² + (EF)² 15² ? 9² + 12² Substitute ? 81 + 144 Multiply 225 = 225 Simplify E 12 F D 15

Slide 11: 

This concludes the Power point show of Lesson 9-5. Return back to lesson.