Bell work What is a circle?

Bell work Answer A circle is a set of all points in a plane that are equidistant from a given point, called the center of the circle A circle is a set of all points in a plane that are equidistant from a given point, called the center of the circle

Unit 3 : Circles: 10.1 Line & Segment Relationships to Circles (Tangents to Circles) Objectives: Students will: 1. Identify Segments and lines related to circles. 2. Use Properties of a tangent to a circle

Words for Circles 1. Concentric 2. Secant 3. Circle 4. Diameter 5. Tangent 6. Arc 7. Chord 8. Radius 9. Interior 10. Semicircle 11. Exterior 12. Congruent 13. Center 14. Point of Tangency Check your answers to see how you did. Are there any words/terms that you are unsure of?

Label Circle Parts 1. Semicircles 2. Center 3. Diameter 4. Radius 9. Tangent 10. Secant 11. Minor Arc 12. Major Arc 5. Exterior 6. Interior 7. Diameter 8. Chord

Lines and Segments related to circles CENTER DIAMETER ALSO A CHORD RADIUS SECANT CHORD TANGENT LINE Exterior Point Interior Point

Lines and Segments related to circles Center of the circle CENTER

Lines and Segments related to circles Diameter – from one point on the circle passing through the center (2 times the radius) DIAMETER ALSO A CHORD CENTER

Lines and Segments related to circles Radius – Segment from the center of the circle to a point on the circle (1/2 the diameter) CENTER RADIUS (I) = 1/2 the Diameter

Lines and Segments related to circles Chord – a segment from one point on the circle to another point on the circle DIAMETER ALSO A CHORD CHORD

Lines and Segments related to circles Secant – a line passing through two points on the circle SECANT

Lines and Segments related to circles Tangent – is a line that intersects the circle at exactly one point TANGENT LINE Point of Tangency

Lines and Segments related to circles CENTER DIAMETER ALSO A CHORD RADIUS SECANT CHORD TANGENT LINE

(p. 597)Theorem 10.1 If a line is tangent to a circle, then it is perpendicular ( _|_ ) to the radius drawn to the point of tangency. If line k is tangent to circle Q at point P, Then line k is _|_ to Segment QP. Q k P Tangent line

Example 1 Find the distance from Q to R, given that line m is tangent to the circle Q at Point P, PR = 4 cm and radius is 3 cm. m 4 cm P 3 cm R Q

Example 1 answer Use the Pythagorean Theorem a² + b² = c² 3² + 4² = c² = c² √25 = √c² √25 = √c² c = 5 c = 5

(p. 597)Theorem 10.2 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. If line k is _|_ to segment QP at point P, then line k is tangent to Circle Q Q k P

Example 2 Given that the radius (r) = 9 in, PR = 12, and QR = 16 in. Is the line m tangent to the circle? m 12 in P 9 in R Q 16 in

Example 2 answer No, it is not tangent. Use the Pythagorean Theorem a² + b² = c² 9² + 12² = 16² = = = 256 Since they are not = then the triangle is not a right triangle and thus the radius is not perpendicular to the line m, therefore the line is not tangent to the circle.

Intersections of Circles No Points of Intersection CONCENTRIC CIRCLES – Coplanar circles that share a common center point

Intersections of Circles One Point of Intersection The Circles are tangent to each other at the point Common Tangents Internal Tangent External Tangent

Intersections of Circles Two Points of Intersection

(p. 598) Theorem 10.3 If two segments from the same exterior point are tangent to a circle, then they are congruent. P P R T __ __ If SR and ST are tangent to circle P, __ SR  ST S

Example 3 Segment SR and Segment ST are tangent to circle P at Points R and T. Find the value of x. P P R T S 2x + 4 3x – 9

Example 3 Answer __ __ __ __ Since SR and ST are tangent to the circle, then the segments are, so circle, then the segments are , so 2x + 4 = 3x – 9 -2x -2x 4 = x – = x

Home work PWS 10.1 A P (10-46) even

Journal Write two things you learned about circles from this section.