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"Symmetry of Circles" Circle PPT Courseware 2
Talk about it
(1) Is a circle an axially symmetrical figure? If so, what is its axis of symmetry? How many axes of symmetry can you find?
(2) How did you come to the conclusion? Communicate with peers.
Basic properties of circles
A circle is an axially symmetrical figure, and its axis of symmetry is any straight line passing through the center of the circle.
Several important concepts
Arc The part between any two points on a circle is called an arc, or arc for short.
Chord A line segment connecting any two points on a circle is called a chord.
Diameter The chord passing through the center of a circle is called diameter.
Note Arcs include superior arcs and minor arcs. Arcs larger than a semicircle are called superior arcs, and arcs smaller than a semicircle are called minor arcs.
Think about it, do it
As shown in the figure, AB is a chord of ⊙O, and the diameter CD is used to make CD⊥AB, and the vertical foot is M.
(1) Is the picture on the right an axially symmetrical figure? If so, what is its axis of symmetry?
(2) What equivalent relationships can you find in the picture? Tell me your reasons.
discovery
It is known that: in ⊙O, the straight line OE passing through the center of the circle is perpendicular to the chord AB, and the vertical foot is E.
Verify: AE=BE, AC=BC, AD=BD.
Proof: Connect OA and OB, then OA=OB. Because the straight line where the diameter CD perpendicular to the chord AB lies is both the symmetry axis of the isosceles triangle OAB and the symmetry axis of ⊙O. Therefore, when the circle is folded along the diameter CD, the two semicircles on both sides of CD coincide with each other, points A and B coincide with each other, AE and BE coincide with each other, and AC and AD coincide with BC and BD respectively.
Therefore AE=BE, AC=BC, AD=BD
Example 1 As shown in the figure, it is known that in ⊙O, the length of chord AB is 8 cm, and the distance from the center O to AB is 3 cm. Find the radius of ⊙O.
Solution: Link OA. Pass O to make OE⊥AB, and the vertical foot is E, then OE=3 cm, AE=BE=1/2AB=4 cm
In RtAOE, according to the Pythagorean theorem, OA = 5 cm
The radius of ∴⊙O is 5 cm.
The converse of the vertical diameter theorem
AB is a chord of ⊙O, and AM=BM.
Draw the diameter CD through point M.
Is the figure on the right an axially symmetrical figure? If so, what is its axis of symmetry?
What equivalence relationships can you find in the picture? Share your thoughts and reasons with your partner.
Xiao Ming found that there are:
From ① CD is the diameter ② CD ⊥ AB, ③ AM = BM ④ AC = BC, ⑤ AD = BD.
The diameter that bisects the chord (not the diameter) is perpendicular to the chord and bisects the two arcs subtended by the chord.
1. Judgment:
⑴A straight line perpendicular to the string bisects the string and bisects the two arcs subtended by the string. ( )
⑵The diameter of an arc subtended by a bisecting chord must bisect the other arc subtended by the chord. ( )
⑶The diameter passing through the midpoint of the string must be perpendicular to the string. ( )
⑷If the arcs included by the two chords of a circle are equal, then the two chords are parallel. ( )
⑸The perpendicular bisector of the string must bisect the arc subtended by the string. ( )
Class summary
1. This lesson mainly studies (1) the axial symmetry of a circle; (2) the perpendicular diameter theorem and its corollary.
2. Questions about chords often require the vertical line segment that passes through the center of the circle to make the chord. This is a very important auxiliary line. The distance from the center of the circle to the chord, the radius, and the length of the chord form a right-angled triangle, and the problem is transformed into a problem of solving the right-angled triangle. .
3. The proof of the vertical diameter theorem is achieved through "experiment-observation-guess-prove", which embodies the practical point of view, the point of view of motion change and the point of view of guessing first and then proving. The introduction of the theorem also applies the method from special to general way of thinking.
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