"Inscribed Circles of Triangles" PPT Courseware 2

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"Inscribed Circles of Triangles" PPT Courseware 2

The picture shows a triangular piece of wood. The carpenter wants to cut a circular piece of material from it. How can he make the area of ​​the cut circle as large as possible?

Construct a circle so that it is tangent to all sides of the given triangle.

Known: △ABC (as shown in the figure).

Find a circle that is tangent to all sides of △ABC.

practice:

1. Draw the bisectors BM and CN of ∠ABC and ∠ACB, and the intersection point is I.

2. Draw ID⊥BC through point I, and draw the vertical foot as D.

3. Taking I as the center of the circle and ID as the radius, make ⊙I. ⊙I is the desired circle.

Positional relationship between triangle and circle

How many such circles can be made? Why?

∵ Straight lines BE and CF have only one intersection point I, and the distances from point I to the three sides of △ABC are equal (why?),

∴Therefore, one and only one circle can be made that is tangent to all three sides of △ABC.

The circle that is tangent to all sides of a triangle is called the incircle of the triangle.

The center of an inscribed circle is called the incenter of a triangle.

This triangle is called the circumscribed triangle of the circle.

Properties of the center of a triangle:

1. The center of a triangle is the intersection of the three angle bisectors of the triangle.

2. The distance from the center of the triangle to each side of the triangle is equal;

Think about it, do it

Question 1: As shown in the figure, in △ABC, point O is the inner center, ∠ABC=50°, ∠ACB=70°, find the degree of ∠BOC.

Variation 1: In △ABC, point O is the inner center, ∠BAC=50°, find the degree of ∠BOC.

Variation 2: In △ABC, point O is the inner center, ∠BOC=120°, find the degree of ∠BAC.

Question 2: Find the radius r of the inscribed circle and the radius R of the circumscribed circle of an equilateral triangle with side length 6cm.

Teacher tips:

First draw a sketch. From the fact that the perpendicular on the base of an isosceles triangle coincides with the bisector of the vertex angle, we know that the inscribed circle and the circumscribed circle of an equilateral triangle are two concentric circles.

Variations:

Find the ratio of the radius r of the inscribed circle to the radius R of the circumscribed circle of an equilateral triangle with side length a.

The "tangent" relationship between triangle and circle

Draw the inscribed circles of right triangles and obtuse triangles respectively, and explain the positions of their centers?

Teacher Tips:

First determine the center and radius of the circle, and keep the traces of the drawing when drawing with a ruler and compass.

Challenge yourself

1. It is known that the lengths of the three sides of △ABC are a, b, and c, and the radius of its inscribed circle is r. Can you find the area of ​​△ABC?

2. It is known that the two right-angled sides of Rt△ABC are a and b respectively. Can you find the radius of its inscribed circle?

fill in the blank:

1. There can be ____ inscribed circles of a triangle, and the incenter of the triangle is on the _______ of the circle.

2. As shown in the figure, O is the center of △ABC, then

OA is divided equally by ∠______, OB is divided equally by ∠______,

OC bisects ∠______,.

(2) If ∠BAC=100º, then ∠BOC=______.

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"Inscribed Circles of Triangles" PPT courseware 3:

"Inscribed Circle of a Triangle" PPT Courseware 3 Review Definition of tangent length: Draw the tangent of the circle through a point outside the circle. The length of the line segment between this point and the tangent point is called the tangent length from this point to the circle. Tangent length theorem: Two tangents to a circle can be drawn from a point outside the circle, and their...

"Inscribed Circles of Triangles" PPT courseware:

"Inscribed Circles of a Triangle" PPT courseware. The picture shows a triangular piece of wood. The carpenter needs to cut a circular piece of material from it. How can the area of ​​the cut circle be made as large as possible? The circle that is tangent to all sides of a triangle is called the incircle of the triangle, and the triangle is called the circumference of the circle.

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Update Time: 2024-11-18

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

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