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Authoritative PPT Summary
"Symmetry of Circles" Circle PPT Courseware 3
learning target
1. Fully grasp the axial symmetry of a circle through the combination of hands and brain.
2. Use exploration and reasoning to fully grasp the perpendicular diameter theorem and its converse theorem in a circle.
3. Expand your thinking, combine it with practice, and use the vertical diameter theorem and its inverse theorem to perform relevant calculations and proofs.
New course introduction
Positional relationship between point and circle
A point is outside the circle. The distance from this point to the center of the circle is greater than the radius.
A point on a circle, the distance from this point to the center of the circle is equal to the radius
A point is inside a circle, and the distance from this point to the center of the circle is less than the radius
(1) Think about it
Is a circle an axially symmetrical figure?
If so, what is its axis of symmetry?
How many axes of symmetry can you find?
Discussion: How did you solve the above problem?
Summary: A circle is an axially symmetrical figure, and its axis of symmetry is any straight line passing through the center of the circle.
(2) Understand arc, chord and diameter
1. Arc: The part between any two points on a circle is called an arc, or arc for short.
As shown in the figure, AB (minor arc), ACD (excellent arc)
2. Chord: A line segment connecting any two points on a circle is called a chord.
As shown in the figure, string AB, string CD
3． Diameter: The chord passing through the center of the circle is called the diameter.
As shown in the figure, diameter CD
Concepts related to circles
An arc larger than a semicircle is called a superior arc.
An arc smaller than a semicircle is called a minor arc
The diameter is the chord, but the chord is not necessarily the diameter;
A semicircle is an arc, but an arc is not necessarily a semicircle;
A semicircle is neither a minor nor a major arc
Do it: follow the steps below
1. Draw an arbitrary ⊙O on a piece of paper, cut out the circle along its circumference, and fold the circle in half so that the two halves of the circle overlap.
2. Get a creased CD.
3． Pick any point A on ⊙O, and draw the perpendicular line of the CD fold through point A to obtain a new fold. Among them, point M is the intersection of the two creases, that is, the vertical foot.
4. Open the paper and the new fold intersects the circle at another point B, as shown in the figure.
question:
(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.
(4) Exploring the converse theorem of the vertical diameter theorem
1. Think about it: As shown in the figure below, AB is the chord (not the diameter) of ⊙O, draw a diameter CD that bisects AB, and intersects AB at point M.
(1) Is this figure 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.
2. Summarize the converse theorem of the vertical diameter theorem:
The diameter that bisects the chord (not the diameter) is perpendicular to the chord and bisects the arc subtended by the chord.
Summary of this lesson
Through the study of this class, we need to master:
1. Related concepts of circles, chords, arcs, superior arcs, and minor arcs.
2. Perpendicular diameter theorem and corollary. And the symmetry of circles.
The diameter perpendicular to the chord bisects the chord and bisects the arc subtended by the chord. The diameter bisects the chord (not the diameter) perpendicular to the chord and bisects the arc subtended by the chord.
Practice in class
1. (2011·Shanghai High School Entrance Examination) As shown in the figure, AB and AC are both chords of the circle O, OM⊥AB, ON⊥AC, and the vertical feet are M and N respectively. If MN=3, then BC=________.
[Analysis] According to the vertical diameter theorem, AN=CN, AM=BM, so BC=2MN=6, answer: 6
2. (2010·Wuhu High School Entrance Examination) As shown in the figure, there is a polyline OABC in ⊙O, where OA=8, AB=12, ∠A=∠B=60°, then the length of BC is ( )
A. 19B. 16C． 18D. 20 Answer:D
3． (2010·Yantai High School Entrance Examination) As shown in the figure, △ ABC is inscribed in ⊙O, D is the midpoint of line segment AB, extended OD intersects ⊙O at point E, and connects AE and BE, then the following five conclusions ①AB⊥DE, ②AE= BE,③OD=DE,④∠AEO=∠C,⑤
The number of correct conclusions is ( )
A.2 B.3 C.4 D.5 Answer: B
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