1. 1. Last year, I received the following email from a former Math 3395 student at KSU: Hello Mr. Koppelman, I am a former student of yours and I have a geometry question for you. I have been playing around with Sketchpad and noticed that if I inscribe a right triangle within a circle with the right angle being at the center of the circle, the ratio of the arc directly opposite of the hypotenuse to the hypotenuse is exactly 1.111. Why!!!????? Using Geometer’s Sketchpad, verify the student’s conjecture. Respond to his question (“Why!!!?????”). r

2. 1. Last year, I received the following email from a former Math 3395 student at KSU: Hello Mr. Koppelman, I am a former student of yours and I have a geometry question for you. I have been playing around with Sketchpad and noticed that if I inscribe a right triangle within a circle with the right angle being at the center of the circle, the ratio of the arc directly opposite of the hypotenuse to the hypotenuse is exactly 1.111. Why!!!????? Using Geometer’s Sketchpad, verify the student’s conjecture. Respond to his question (“Why!!!?????”). Circumference = 2πr r

3. 2. Using Geometer’s Sketchpad, construct a circle and a random point P outside the circle (as shown). Construct a line through P that is tangent to the circle. Your line should remain a tangent line no matter where point P is moved outside the circle.

4. PA = 9 PC cannot be longer than tangent segment PF

5. D B C A

6. D B C A x x 2x x+90 ½ (2x + 180) 180  The measure of angle C exceeds the measure of angle D by 90 .

7. 100 π

8. 3

9. Q(a,0) P

10. P 4 a 1 3 

12. x 6 6 6 – x 3y3y y In  BPD 6 2 + x 2 = (3y 2 )  36 + x 2 = 9y 2

14. B R P A Q C

15. B R P A Q C Outline of Proof The measures of the 3 angles marked in RED are congruent. Therefore, quadrilateral BRQC is cyclic Since  BRQ is supplementary to  PRQ, then  BRQ and  C are supplementary.      Thus the opposite angles of quadrilateral BRQC are supplementary.

16. 13  54  41 

17. 13  Extend radius PC through P to meet the circle again at point Q. Draw QB. <CBQ is a right angle (inscribed in a semicircle) and m<BQC = m<CAB = 54. Therefore,  BCQ is similar to  DCA, and m<QCB = m<ACD = 36. Since m<ACB = 85, m<DCP = 85 – 72 = 13°. 54  41  Q

18. 12. In the diagram, radius PA has length 7 inches and the length of segment BC is 8 inches. The measures of angles APB and PBC are each 60 degrees. There are two possible values for the length of segment PB. Find both. C ½ x B P 60  7 7 8 x 7– x y  A (7 – x)(7 + x) = 8y y + ½ x = 8 – ½ x  y = 8 – x Substituting: 49 – x 2 = 8(8 – x)  49 – x 2 = 8y  x = 5 or 3 49 – x 2 = 64 – 8x x 2 + 8x – 15 = 0 (x – 5)(x – 3) = 0 3, 5

19. The circle is circumscribed around  ABC  ABC is inscribed in the circle The circumcenter of a triangle is the center of the circumscribed circle. It is the intersection of the perpendicular bisectors of the sides of the triangle. The circumcenter of a triangle is equidistant from the vertices of the triangle.

20. The incenter of a triangle is the center of the inscribed circle. It is the intersection of the angle bisectors of the angles of the triangle. The incenter of a triangle is equidistant from the sides of the triangle. The circle is inscribed in  ABC  ABC is circumscribed around the circle

21. The orthocenter of a triangle is the intersection of the altitudes.

22. The centroid of the triangle is the intersection of the medians.

23. The 3 altitudes of  ABC O orthocenter O

24. The three perpendicular bisectors of  ABC O P orthocenter O circumcenter P

25. The three medians of  ABC O P R circumcenter P orthocenter O The Euler Line centroid R

26. O P R circumcenter P orthocenter O The Euler Line centroid R     The Nine Point Circle

27. Euler Line Midpoint of segment from orthocenter to incenter (center of nine point circle) The Nine Point Circle Circumcenter

28. Conic Sections are formed by the intersection of a plane and a right circular cone. The type of curve produced is determined by the angle at which the plane intersects the cone.

29. A parabola is the locus (set) of points that are the same distance from a fixed point (called the focus) and a fixed line (called the directrix).  P l  

30.  P l   

31.  P l       Focus Directrix   

32. Construction steps: 1.Construct line l and point P not on l. 2.Choose a point Q on line l. 3.Construct segment PQ. 4.Construct the perpendicular bisector of PQ. 5.Construct the perpendicular to line l through point Q and label its intersection with the perpendicular bisector, point A. 6.Trace point A. 7.Animate point Q. 8.Point A will trace a parabola. l A parabola is the locus (set) of points that are the same distance from a fixed point (called the focus) and a fixed line (called the directrix). Q P Q A  o  o  o  o

33. An ellipse is the locus (set) of points the sum of whose distances from two fixed points (called foci) is constant.    12 88

34. An ellipse is the locus (set) of points the sum of whose distances from two fixed points (called foci) is constant.    12 

35. An ellipse is the locus (set) of points the sum of whose distances from two fixed points (called foci) is constant.    12  2 

36. An ellipse is the locus (set) of points the sum of whose distances from two fixed points (called foci) is constant.    12    115 

37. An ellipse is the locus (set) of points the sum of whose distances from two fixed points (called foci) is constant.    12    11 5   

38. An ellipse is the locus (set) of points the sum of whose distances from two fixed points (called foci) is constant.    12       

39. Construction steps: 1.Construct a circle with center A. 2.Choose point P on the circle. 3.Construct radius PA. 4.Choose point B inside the circle. 5.Construct line segment PB. 6.Construct the perpendicular bisector of PB through midpoint M. 7.Label the intersection of PA and the perpendicular bisector, point E. 8.Display the lengths of AE and BE, and their sum. 9.Trace point E 10. Animate point P. 11. Point E will trace an ellipse with foci A and B. An ellipse is the locus (set) of points the sum of whose distances from two fixed points (called foci) is constant.

40. A hyperbola is the locus (set) of points the difference of whose distances from two fixed points (called foci) is constant.

41. Construction Steps: 1. Draw a circle and hide the point on the circle. Label the center of the circle A. 2. Construct a new point on the circle. Label it P. 3. Draw a point outside of the circle and label it B. 4. Construct the line PA. 5. Construct the segment PB. 6. Construct the perpendicular bisector of PB to form point M. 7. Construct the intersection of the line PA and the perpendicular bisector of PB and label it H. 8. Trace point H as you move point P around the circle. Outline of proof that point H traces out a hyperbola: 1. HM  PB by construction. 2.  HMP and  HMB are congruent right angles. 3. MP  MB by the definition of bisector. 4. HM  HM by the reflexive property. 5.  PMH   BMH by SAS. 6. PH  BH by CPCTC. 7. PH – AH = PA which is the radius of the circle (a constant). 8. Since PH  BH, then BH – AH = the radius (a constant) by substitution. 9. Therefore, Point H traces out a hyperbola since the difference of the distances (BH – AH) from the foci (A and B) is constant. A hyperbola is the locus (set) of points the difference of whose distances from two fixed points (called foci) is constant.

42. Homework #18 Prove that the construction of the ellipse works.