"Symmetry of Circles" PPT courseware

Description

"Symmetry of Circles" PPT courseware

Class objectives

1. Experience the process of exploring the symmetry and related properties of circles.

2. Understand the symmetry and related properties of circles.

3． Use the vertical diameter theorem to solve related problems.

Review questions:

1. What is an axially symmetrical figure? What axisymmetric shapes have we learned about straight lines?

If a figure is folded in half along a straight line and the parts on both sides of the straight line overlap each other, then the figure is called an axially symmetrical figure. Such as line segments, angles, isosceles triangles, rectangles, rhombuses, isosceles trapezoids, squares

Is a circle an axially symmetrical figure?

If so, what is its axis of symmetry? How many axes of symmetry can you find?

How did you solve the above problem?

A circle is an axially symmetrical figure.

The axis of symmetry of a circle is any straight line passing through the center of the circle. It has countless axes of symmetry.

The above problems can be solved by using the folding method.

Exercise 1. True or False Questions

(1) The diameter is the chord. (√) (2) The line segment passing through the center of the circle is the diameter. (×)

(3) A semicircle is an arc. (√) (4) Two semicircles are equal arcs. (×)

(5) Two circles with unequal areas are not equal circles. (√)

(6) Two arcs of equal length are equal arcs. (×)

Discuss

AB is a chord of ⊙O.

Let CD⊥AB be the diameter, and the vertical foot is M.

Is the figure below 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.

topic conclusion

From ①CD is the diameter ③AM=BM,

② CD⊥AB ④AC=BC,

⑤AD=BD.

Perpendicular diameter theorem

The diameter perpendicular to the chord bisects the chord and bisects the two arcs subtended by the chord.

Perpendicular diameter theorem in three languages

Theorem: A diameter perpendicular to a chord bisects the chord and bisects the two arcs subtended by the chord.

Lectures with classic examples

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: Connect OA. Pass O to make OE⊥AB, and the vertical foot is E.

Then OE=3 cm, AE=BE.

∵AB=8cm ∴AE=4cm

In Rt △AOE, according to the Pythagorean theorem, OA = 5 cm

The radius of ∴⊙O is 5 cm

inference

The diameter that bisects the chord (not the diameter) is perpendicular to the chord and bisects the two arcs subtended by the chord.

Any two diameters of a circle always bisect each other, but they are not necessarily perpendicular to each other. Therefore, if the string here is a diameter, the conclusion may not be true.

Challenge yourself to draw a picture

As shown in the figure, M is a point within ⊙O. Use a ruler and compass to draw a chord AB so that AB passes through point M. And AM=BM.

Comply with the standard in class. Fill in the form

⑴A straight line perpendicular to the string bisects the string and bisects the arc subtended by the string ( )

⑵The line connecting the midpoints of the two arcs opposite to the chord is perpendicular to the chord and passes through the center of the circle ( )

⑶Chords of a circle that are not perpendicular to the diameter must not be bisected by the diameter ( )

⑷The diameter that bisects the chord is perpendicular to the chord and bisects the two arcs subtended by the chord ( )

⑸Two non-diameter chords in a circle cannot bisect each other ( )

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Update Time: 2024-10-05

This template belongs to Mathematics courseware Qingdao Edition Ninth Grade Mathematics Volume 1 industry PPT template

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