Geometry  Semester 2. Mrs. DayBlattner 1/20/2016


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1 Geometry  Semester 2 Mrs. DayBlattner 1/20/2016
2 Agenda 1/20/2016 1) 20 Question Quiz  20 minutes 2) Jan 15 homework  selfcorrections 3) Spot check sheet Thales Theorem  add to your response 4) Finding the center of circles. Introductory activity. 5) Homework
3 Problem Set  correct your own work, use color B A.O C 1. A, B, and C are three points on a circle, and angle ABC is a right angle. What s wrong with the picture? Explain your reasoning. Student s said angle ABC couldn t be a right angle because AC wasn t a diameter of the circle. That is correct.
4 Problem Set  correct your own work, use color to B show this proof also A.O C Draw in the 3 radii from O to A, B and C.  isosceles triangles, label congruent base angles (a, b, and c ). For ABC
5 Problem Set  correct your own work, use color B A.O C Draw in the 3 radii from O to A, B and C.  isosceles triangles, label congruent base angles (a, b, and c ). For ABC 2a + 2b + 2c = 180
6 BUT, angle B = 90 = b + c Problem Set  correct your own work, use color B A.O C For ABC 2a + 2b + 2c = 180 2(a + b + c ) = 180 a + b + c = 90
7 Problem Set  correct your own work, use color B A.O C a + b + c = 90 and 90 = b + c can t both be true at the same time  we have a contradiction (a can t be 0 )
8 since line segment AC is not a diameter of the circle. Also, can do a proof similar to answer for Show that there is something mathematically wrong with the picture below. Again students noted that angle ABC couldn t be a right angle, B C A.O
9 3. In the figure, AB is the diameter of a circle of radius 17 miles. If BC = 30 miles, what is AC? C 30 miles Use Pythagorean theorem AC 2 + (30mile) 2 = (34miles) 2 AC = sq.rt [(34mile) 2 (30miles) 2 ] AC = sq.rt [( ) miles 2 ] A 17 miles. 17 miles B AC = sq.rt256 miles AC = 16 miles
10 4. In the figure below, O is the center of the circle, and AD is a diameter. a) Find measure of angle AOB = 48 isosceles triangle BOD, angle BOD = 180 angle BOD = 132 Angle AOB and BOD are linear pairs, so Angle AOB = 48 b) If measure angle AOB: measure of angle COD = 3: 4, what is measure of angle BOC? m angle AOB
11 4. In the figure below, O is the center of the circle, and AD is a diameter. b)if measure angle AOB: measure of angle COD = 3: 4, what is measure of angle BOC? 48 = 3 m COD 4 m COD = 4(48 ) / 3 = 64 m BOC = = 68
12 5. PQ is a diameter of a circle, and M is another point on the circle. The point R lies on the line MQ such that RM = MQ. Show that measure of angle PRM = measure of angle PQM. HINT you need to draw a picture to help you explain the situation.
13 Label the lengths that you know are congruent, and the right angles and write a proof to show m PRM = m PQM M Q center P O
14 6. Inscribe triangle ABC in a circle of diameter 1 such that AC is a diameter. Explain why: a) sin(angle A ) = BC sin (angle A) = side opposite angle A / hypotenuse = BC /1 = BC b) cos(angle A) = AB cos(angle A) = side adjacent to angle A / hypotenuse = AB / 1 = AB
15 Spot Check: Thales Theorem The shape defined by the endpoints of the two diameters will always form a rectangle. According to Thales theorem, whenever an angle is drawn from the diameter of a circle to a point on its circumference, then the angle formed is a right angle. All four endpoints represent angles drawn from the diameter of the circle to a point on the cirumference, therefore each of the four angles is a right angle.
16 Spot Check: Thales Theorem The shape defined by the endpoints of the two diameters will always form a rectangle. A quadrilateral with 4 right angles, will be a rectangle by definition of a rectangle. (A square is ALSO a rectangle, but one with additional properties. When the diameters cross at right angles then you will draw a square.)
17 Homework  Finding the Center On colored paper (or white unlined paper) plot 3 points (that are not all in a line). Label them (e.g. A, B, C). Then draw line segments that connect pairs of points. Use your compass and straightedge to construct the perpendicular bisectors of the line segments. Place your compass point where 2 of the perpendicular bisectors intersect, then stretch out the pencil end of the compass to any one of your points (A, B or C) and draw a circle. You should find that you have found the point that is the same distance from all 3 points and so the circle you draw will have A, B and C all as points on the circumference of that circle. Do this activity 3 times to check that the process works every time and to help you gain confidence and skill doing the construction. Bring your papers with you to class on Friday  I will be collecting this homework assignment. See me at enrichment on Thursday if you need help. I'll be in room the computer lab.
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