Unit 3 Lesson 2 Circle Property

Transcription

1 Unit 3 Lesson 2 ircle Property What are we going to learn in this lesson? What is a chord? The 2 chord properties. n introduction to proofs with circle. The relationship between an inscribed angle and a central angle. The relationship between two inscribed angles. The relationship of interior angles in a cyclic quadrilateral. The relationship between exterior angles and opposite interior angles. The relationship between the two angles subtended by the same arc.

2 What is a chord? chord is line segment that joins two points on the circle. diameter is a chord that passes through the centre of the circle. secant is a line that contains a chord. In application: Use the above definition to draw a chord, diameter and a secant in the following circle. secant diameter chord hord Properties (1) hord Perpendicular isector Theorem (p. 398) The perpendicular bisector of a chord contains the centre circle. The perpendicular bisectors of two non parallel chords intersect at the centre of the circle. The perpendicular from the centre of a circle to a chord bisects the chord. The line segment joining the centre of a circle and the midpoint of a chord is perpendicular to the chord. In application: raw 2 diagrams that show the hord Perpendicular isector Theorem.

3 hord Property (2) ongruent hords Theorem (p. 399) If two chords are congruent, they are equidistant from the centre of a circle. The converse of this theorem is also true. If two chords are equidistant from the centre of a circle, the chords are congruent. In application: raw a diagram that shows the ongruent hords Theorem. Example 1. Find the value of y. 5 y 6

4 Example 2. Given: Find the length of 4 E 3 F 3 Practice 1. Find the area of the triangle given that E

5 Practice 2. etermine the value of x. Worksheet p. 400 #s 5 19

6 Page 400 # a) 5.8 cm b) 10 cm cm 7. a) 1 cm b) 5.3 cm 8. a) 10.8 cm b) 14.8 cm 9. a) 14.2 cm b) 22 cm 10. a) 8.5 cm b) 17 cm 11. a) 5.2 cm b) 10.4 cm 12. a) 6.4 cm b) 2.4 cm cm 14. a) 4.0 cm b) 4.5 cm o 16. a) 48 o b) 49 o cm cm cm ther examples (proofs) 1. Given: XY, XZ, and YZ are equidistant from the centre of a circle Prove : is equilateral. X Z Y

7 ther examples (proofs) 2. Given: is a chord of a circle with centre E Prove: Worksheet p. 400 #s 20, 21

8 Page 400 #20 Given Given efinition of perpendicular hord property ommon SS Two corresponding sides of congruent triangles ITT E Page 400 #21 Given Given efinition of midpoint ommon Radius of circle SSS Two corresponding angles of congruent triangles Supplementary angles E

9 alculate the value of s s 10 What is an inscribed angle and what is a central angle? n inscribed angle has its vertex on the circle, and two chords form the arms. central angle has its vertex at the centre of a circle, and two radii form the arms. In application: Use the above definition to draw an inscribed angle and a central angle in the following circle. inscribed angle central angle

10 Inscribed ngles Properties (1) ngles in a ircle Theorem, Part 1 The measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc. In application: Use the above definition to represent the ngles in a ircle Theorem, Part 1 in the following circle. Inscribed ngles Properties (2) ngles in a ircle Theorem, part 2 Inscribed angles subtended by the same arc are congruent. Inscribed angles subtended by equal arcs are congruent. In application: Use the above definitions to represent each part of the ngles in a ircle Theorem, part 2 in the following circles.

11 Example (p. 411, Example 4) 1. etermine the measures of angles 1 and o o E Example (p. 412, Example 5) 2. The centre of the circle is. etermine the measures of and. 56 o

12 Practice 1. etermine the measure of angles 1, 2 and o 2 3 Practice 2. In the following diagram, mesures 80 o. etermine the measures of and.

13 Worksheet p #s 5 24 Page # o o o o o o 11. a = 50 o b = 100 o 12. a = 20 o b = 140 o c = 80 o 13. a = 48 o b = 42 o c = 90 o 14. x = 110 o 15. a = 40 o b = 40 o 16. a = 99 o b = 57 o 17. a = 61 o b = 29 o c = 29 o 18. a = 40 o b = 40 o 19. a = 56 o b = 112 o c = 68 o d = 34 o 20. t = 20 o x = 50 o y = 70 o z = 70 o 21. x = 40 o y = 25 o z = 60 o 22. x = 45 o y = 35 o z = 45 o 23. a = 40 o b = 30 o c = 70 o d = 40 o e = 40 o f = 70 o 24. w = 40 o x = 80 o y = 40 o z = 50 o

14 ther Example (p. 413 # 26) 1 2 E 3 4 Given Inscribed ngles Properties Inscribed ngles Properties "Z" Transitivity ITT ITT (onverse) ITT (onverse) ddition Property Page 413 # 25 Given Inscribed ngle Properties Inscribed ngle Properties ommon S E

15 What is a yclic Quadrilateral? set of points is called cyclic (or concyclic) if all of the points lie on the same circle. polygon with all its vertices on the same circle is called a cyclic polygon. Then, a cyclic quadrilateral is a polygon with 4 sides with all its vertices on the same circle. In application: Use the above definition to draw a cyclic quadrilateral in the following circle. an you determine the relationship between the interior angles? yclic Quadrilaterals properties (1) yclic Quadrilateral Theorem, part 1 The opposite angles of a cyclic quadrilateral are supplementary. In application: Use the above definition to represent the yclic Quadrilateral Theorem part 1 in the following circle. a d a + c = 180 o b c b + d = 180 o

16 yclic Quadrilaterals properties (2) yclic Quadrilateral Theorem, part 2 n exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. In application: Use the above definition to represent the yclic Quadrilateral Theorem part 2 in the following circle. yclic Quadrilaterals properties (3) yclic Quadrilateral Theorem, part 3 The line segment joining two vertices of a cyclic quadrilateral subtends equal angles at the other two vertices on the same side of the segment. In application: Use the above definition to represent the yclic Quadrilateral Theorem part 2 in the following circle.

17 Example etermine the measures of the following angles. Worksheet p #s 1 10