Geometry - Semester 2. Mrs. Day-Blattner 1/20/2016

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1 Geometry - Semester 2 Mrs. Day-Blattner 1/20/2016

2 Agenda 1/20/2016 1) 20 Question Quiz - 20 minutes 2) Jan 15 homework - self-corrections 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 straight-edge 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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