"Determining the Conditions of a Circle" Circle PPT Courseware 3

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"Determining the Conditions of a Circle" Circle PPT Courseware 3

Knowledge review

1. The circumferential angle subtended by the diameter is a right angle;

2.The chord subtended by a circumferential angle of 90° is the diameter.

3. The four vertices of the quadrilateral ABCD are all on ⊙O. Such a quadrilateral is called a circle inscribed in the quadrilateral; this circle is called the circumcircle of the quadrilateral.

4. The diagonals of a quadrilateral inscribed in a circle are complementary.

Can a circle be determined through three known points A, B, and C?

Assume that ⊙O exists through three points A, B, and C

(1) The distance from the center O to the three points A, B, and C is ______ (fill in “equal” or “unequal”).

(2) Connect AB and AC, and draw straight lines MN⊥AB and EF⊥AC respectively through point O, then MN is the ______ of AB; EF is the ______ of AC.

(3) The distance from the intersection point O of the perpendiculars of AB and AC to B and C is ______.

Three points determine the circle

Theorem Three points that are not on the same straight line determine a circle.

During the drawing process above.

∵ Lines DE and FG have only one intersection point O, and the distances from point O to three points A, B, and C are equal.

∴A circle can be drawn through three points A, B, and C, and only one circle can be drawn.

Please draw a circle so that it passes through the known points A, B, and C (A, B, and C are not collinear).

Method: 1. Connect AB and draw the vertical bisector MN of line segment AB;

2. Connect AC, draw the vertical bisector EF of line segment AC, and intersect MN at point O;

So point O is the point you want to construct.

Positional relationship between triangle and circle

Therefore, the three vertices of a triangle define a circle, which is called the circumcircle of the triangle. This triangle is called the inscribed triangle of the circle.

The center of the circumscribed circle is the intersection point of the perpendicular bisectors of the three sides of the triangle, which is called the circumcenter of the triangle.

Teacher Tips:

The positional relationship between the vertices of a polygon and a circle is called a connection.

1. The circumcenter of the triangle is ()

A. The intersection point of the three midlines B. The intersection point of the mid-perpendicular lines of the three sides

C. The intersection of three heights D. The intersection of three angle bisectors

2. The circumcenter of an acute triangle is located inside the triangle.

The circumcenter of a right triangle is at the midpoint of the hypotenuse.

The circumcenter of an obtuse triangle is outside the triangle.

Knowledge and skills:

1. There are three grazing points on the grassland. It is necessary to build a herdsman settlement so that the distances from the three grazing points to the settlement are equal. If the positions of the three grazing points are as shown in the picture, how to determine the location of the settlement?

Method: 1. Connect AB and draw the vertical bisector MN of line segment AB;

2. Connect AC, draw the vertical bisector EF of line segment AC, and intersect MN at point O;

So point O is the point you want to construct.

practice

1. Which of the following propositions is incorrect ( )

A. There are countless circles passing through one point. B. There are countless circles passing through two points.

C. A circle can be determined by passing through three points. D. It cannot be determined by passing through three points on the same straight line.

2. The properties of the circumcenter of a triangle are ( )

A. The distance to the three sides is equal. B. The distance to the three vertices is equal.

C. The circumcenter is outside the triangle. D. The circumcenter is inside the triangle.

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"Conditions for Determining a Circle" PPT courseware learning objectives 1. Knowledge and skills: ① Understand how three points that are not on the same straight line determine a circle; ② Master the method of making a circle with three points that are not on the same straight line; ③ Understand the principles of triangles Concepts such as circumcircle and circumcenter of triangle...

"Determining the Conditions of a Circle" Circle PPT Courseware 5:

"Determining the Conditions of a Circle" Circle PPT Courseware 5 Knowledge review: 1. The method and properties of perpendicular bisectors of line segments 2. How many straight lines can be drawn through a point? 3. At what point can a straight line be determined? Exploration 1 Can a circle be determined through a known point A? Exploration 2 Passed...

"Determining the Conditions of a Circle" Circle PPT Courseware 4:

"Conditions for Determining a Circle" Circle PPT Courseware 4 Learning Objectives 1. Understand how to determine a circle from three points that are not on the same straight line, and how to draw a circle through three points that are not on the same straight line. 2. Understand the circumcircle of a triangle, Concepts such as the circumcenter of a triangle. 3. Experiences are not the same..

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Update Time: 2024-09-22

This template belongs to Mathematics courseware Beijing Normal University Ninth Grade Mathematics Volume 2 industry PPT template

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