"Symmetry of Circles" Circle PPT Courseware

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"Symmetry of Circles" Circle PPT Courseware

Perpendicular diameter theorem in three languages

As shown in the figure ∵ CD is the diameter, CD⊥AB,

∴AM=BM,AC=BC,AD=BD.

The converse of the vertical diameter theorem

As shown in the figure, under the following five conditions:

① CD is the diameter (through the center of the circle), ② CD⊥AB, ③ AM=BM,

④AC=BC,⑤AD=BD.

As long as two of the conditions are met, the remaining three conclusions can be derived.

judge:

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 this 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. ( )

Example 1, as shown in the figure, in ⊙O, CD is the diameter, AB is the chord, and CD⊥AB, it is known that CD = 20, CM = 4, find AB.

Note: When solving similar problems, we often make OM and AO first, and then use the vertical diameter theorem and the Pythagorean theorem.

Perpendicular diameter theorem triangle

Known: As shown in the figure, the diameter CD⊥AB, the vertical foot is E.

⑴If the radius R = 2 and AB =2√3, find the lengths of OE and DE.

⑵If the radius R = 2 and OE = 1, find the lengths of AB and DE.

⑶Inspired by questions ⑴ and ⑵, what other questions can you make up?

Example 2: The radius of ⊙O is 10, the chord AB=12, CD=16, and AB∥CD. Find the distance between AB and CD.

Analysis: This question is an "unclear graphic" question and should be solved by classification. (As shown on the right)

Example 3: As shown in the figure, P is a point outside ⊙O, and the rays PAB and PCD intersect ⊙O at A, B, C, and D respectively. It is known that AB=CD,

Prove: PO equals ∠BPD

Extension: Can a ship cross an arch bridge?

As shown in the picture, there is an arc-shaped arch bridge in a certain place. The width of the water under the bridge is 7.2 meters, and the arch is 2.4 meters above the water. There is an existing cargo ship with a width of 3 meters, a rectangular top of the cabin and a height of 2 meters above the water. It needs to pass here. Can this cargo ship successfully pass through this arch bridge?

Class exercises

In ⊙O, if CD ⊥AB is in M, and AB is the diameter, then which of the following conclusions is incorrect ( )

A. AC=AD B. BC=BD

C. AM=OM D. CM=DM

It is known that the diameter of ⊙O is AB=10, the chord CD is ⊥AB, the vertical foot is M, and OM=3, then CD=____.

In ⊙O, CD ⊥AB is in M, and AB is the diameter. If CD=10, AM=1, then the radius of ⊙O is ____.

Class summary: In this lesson, we explored and discovered Corollary 1 and Corollary of the Perpendicular Diameter Theorem.

Argument 2, and use Corollary 1 to bisect the arc.

●It is necessary to distinguish the question and conclusion of Corollary 1, that is, what conditions are known and what conclusions can be derived. This is the key to correctly understanding the application of Corollary 1;

●Example 3 is a basic geometric drawing. You will bisect the arc by making the vertical bisector of the chord between the arc. You can understand the application of the transformation idea here.

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Update Time: 2024-09-25

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