1. Spi.3.1 Use algebra and coordinate geometry to analyze and solve problems about geometric figures (including circles). Check.2.2 Approximate pi from a table of values for the circumference and diameter of circles using various methods (e.g. line of best fit). Check.4.39 Identify lines and line segments associated with circles. Spi.4.8 Solve problems involving area, circumference, area of a sector, and/or arclength of a circle. Spi.4.13 Identify, analyze and/or use basic properties and theorems of circles to solve problems (including those relating right triangles and circles). 10.5 Tangents

2. Constructions, page 720 and lab on Page 726

3. Constructions

4. 10.5 Tangents Action is the foundational key to all success. Pablo Picasso Objective: Identify tangents and solve problems involving circumscribed polygons O R T If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Converse: If a line is perpendicular to the radius at its endpoint, then the line is tangent to a circle. D A B C If two segments from the same exterior point are tangent to a circle, then they are congruent. AB  BC

5. Find Lengths ED is tangent to circle F at point E. Find x. Tangent meets at a right angle x 2 = 4 2 + 3 2 x = 5

6. Find Lengths SR is tangent to circle Q at point R. Find y. Tangent meets at a right angle 20 2 = 16 2 + x 2 400 – 256 = 144 = x 2 X=12 Y = 24

7. Determine if MN is tangent to circle L Is  LMN a right triangle? LN = LO + ON LN = 3 + 2 = 5 5 2 = 4 2 + 3 2 Yes MN is tangent Determine is PQ is tangent to circle R RP = RS + SP = 8 8 2 = 4 2 + 5 2 64 = 16 + 25 64  41 No it is not tangent

8. Find X, assume AD, AC and AB are tangents AB  AC and AC  AD AB  AD reflexive property -2x + 37 = 6x + 5 32 = 8x 4 = x AB = -2(4) + 37 = 29

9. Circumscribed Polygons

10. Triangle ADC is circumscribed about circle O. Find the perimeter of ADC if EC = DE + AF DE = 6 FA = 19 BC = EC = 6 + 19 = 25 Perimeter = 2(6) + 2(19) + 2(25) = 12 + 38 + 50 = 100

11. Circumscribed Polygons Triangle HJK is circumscribed about circle G. Find the perimeter of HJK if NK = JL + 29 JL = 16 HN = 18 MK = NK = 16+ 29 = 45 Perimeter = 2(16) + 2(18) + 2(45) = 158

12. Tangents Summary Practice Assignment: Page 722, 8 - 24 even* Honors Page 722, 12 – 24 even* and 30 O R T If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Converse: If a line is perpendicular to the radius at its endpoint, then the line is tangent to a circle. D A B C If two segments from the same exterior point are tangent to a circle, then they are congruent. AB  BC