"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 2

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"The Relationship between Circumferential Angle and Central Angle" Circle PPT Courseware 2

1. Review

1.What is the angle of a circle?

The vertex is on the circle and the angle where the two sides intersect the circle is called the circumferential angle.

2. Fill in the blanks:

⑴The _______ subtended by an arc is equal to half of the degree __________ it subtends.

⑵The measure of a circumferential angle is equal to the arc it subtends __________.

⑶The central angle subtended by an arc is equal to the _______ of the circumferential angle subtended by it.

3. Angle and arc are closely related, so when proving the relationship of angles, you can consider proving the relationship of the arc to which the angle corresponds.

4. The proof of the circle angle theorem applies the classification idea in mathematics

Circle Angle Theorem:

The circumferential angle subtended by an arc is equal to half of the central angle subtended by it.

Corollary 1: The diagonals of a quadrilateral inscribed in a circle are complementary.

A quadrilateral with complementary diagonals is inscribed in a circle.

Corollary 2: The circumferential angles subtended by the same arc or equal arcs are equal;

In congruent or congruent circles, equal arcs subtended by equal circumferential angles are also equal.

Corollary 3 The circumferential angle subtended by the semicircle (or diameter) is a right angle;

The chord subtended by a 90-degree circumferential angle is the diameter.

Corollary 4 If the midline on one side of a triangle is equal to half of this side,

Then this triangle is a right triangle.

Example 1. (Beijing High School Entrance Examination Questions in 1999)

In ⊙O, CD passes through the circle center O, and CD⊥AB is at D. Let's draw a chord CF through the point C to intersect ⊙O at F and AB at E. Prove: CB²=CE·CF

As shown in the figure, the diameter of AE⊙O, the vertices of △ABC are all on ⊙O, and AD is the height of △ABC; prove: AB · AC = AE · AD

Variation: ⑴ △ABC is inscribed in ⊙O, if ∠1=∠2. Prove: AB•AC=AE•AD.

Variation: ⑵△ABC is inscribed in ⊙O, if the string AE bisects ∠BAC. Prove: AB•AC=AE•AD.

Example 2: It is known that as shown in the figure, the circle is inscribed in △ABC, AB=AC, and the chord AE intersects BC at D

Prove: ⑴△ABD∽△AEB ⑵AB²=AD·AE

4. Summary

In this class, we learned the circle angle theorem and three corollaries:

Corollary 1. A quadrilateral inscribed in a circle has complementary diagonals; a quadrilateral with complementary diagonals is inscribed in a circle.

Corollary 2. The circumferential angles subtended by the same arc or equal arcs are equal; the arcs subtended by equal circumferential angles in the same circle or equal circles are equal.

Corollary 3. The circumferential angle subtended by the semicircle (or diameter) is a right angle; the chord subtended by the 90º circumferential angle is the diameter.

Corollary 4. If the midline of one side of a triangle is equal to half of that side, then the triangle is a right triangle.

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Update Time: 2024-11-17

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