Lesson 2B: Thales Theorem
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1 Lesson 2B: Thales Theorem Learning Targets o I can identify radius, diameter, chords, central circles, inscribed circles and semicircles o I can explain that an ABC is a right triangle, then A, B, and C are three distinct points on a circle with a diameter AB. Opening Exercise a. Draw any Radius in the circle 0. Label the endpoint as A. b. What do you know about radii? Home many radii a circle have? O c. Extend radius OA to be a Diameter. Label the new end point as B What do you know about Diameters? How many diameter a circle have? d. Draw another radius. This time, label the endpoint as C. What do you notice about < AOC? e. Connect A to C to create AC. This segment is called Chord f. Highlight BAC, this is called the Inscribe angle
2 Draw a diagram for each of the vocabulary words Definition Diagram Circle: The set of all points from a given point Radius: A that connects the of the circle to any point on the circle Diameter: A that passes through the and whose are on the circle Chord: A whose are on the circle Central Angle: An whose vertex is on the of the circle Inscribed Angle: An angle whose is on a circle and each of the angle intersects the circle in another. Semicircle: a circle formed by a
3 Example 1. a. Two points A and B are given. Take the index card and push it up between points A and B. b. Mark on your paper the location of the corner of the colored index card and label this as point C.(make sure the sides of the index card are always touching A and B) c. Do this again, pushing the corner of the colored index card up between A and B but at a different angle. Again, mark the location of the corner, labeling it as point D. A C B d. Continue locating points in the same manner in all directions through A and B, labeling the points as you go (create at least 8 eight points). e. What shape do the points create? f. Connect points A and B. What have you created? g. Draw in ACB. What type of angle is ACB? h. What type of triangle is ACB? THALES THEOREM: If A, B, and C are three different points on a circle with a diameter AB, then ACB is a right angle. In other words - Two chords and a diameter will always create a right triangles in a circle. Inscribed angles in a semicircle are always right angles
4 Example 2 Circle O is shown below. a. Draw diameter AB and CD of the circle. b. Connect the endpoints of the diameters c. What quadrilateral is ABCD? Explain your answer Example 3. AB is a diameter of the circle shown. The radius is cm, and AC = 7 cm. a) Find m C. b) Find AB. c) Find BC. Example 4. In the circle shown, BC is a diameter with centera. (Hint: What do you know about all the radii of a circle) Find m DAB. Find m BAE. Find m DAE.
5 Lesson 2B: Thales Theorem Classwork 1. * Give an example of each of the following for the given diagram o Diameter o Radius o Chord o Central angle o Inscribed angle o Right angle 2. **Determine the length of the radius of the circumscribed circle To the right triangle with legs 7 cm and 4 cm. Round your answer to the nearest hundredth 3. A, B, and C are three points on a circle, and angle ABC is a right angle. What is wrong with the picture below? Explain your reasoning.
6 4. ** Explain why there is something mathematically wrong with the picture at right. 5. **In the figure below AB is the diameter of the circle with radius 17 miles. miles. If BC = 30 miles, what is AC? 6. ****In the figure below, O is the center of the circle, and AD is a diameter. a. Find m AOB. b. If m AOB m COD = 3 4, what is m BOC? 7. ***Inscribe ABC in a circle of diameter 1 such that AC is a diameter. Explain why: a. sin( A) = BC. b. cos( A) = AB.
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