Topicm37c0e2b077cc18f8_1528449000663_0Topic

The circle inscribed in a triangle

Levelm37c0e2b077cc18f8_1528449084556_0Level

Second

Core curriculumm37c0e2b077cc18f8_1528449076687_0Core curriculum

XV. Symmetries. The student:

1) identifies the perpendicular bisector and the angle bisectorangle bisectorangle bisector of line segments;

2) knows and uses practically the basic properties of the perpendicular bisector and the angle bisectorangle bisectorangle bisector of the line segment in the sample exercise below: 
The vertex C of an ABCD rhombus is located on the perpendicular bisectors of sides AB and AD. Calculate the angles of this rhombus.

Timingm37c0e2b077cc18f8_1528449068082_0Timing

45 minutes

General objectivem37c0e2b077cc18f8_1528449523725_0General objective

Using simple, well known mathematical objects, interpreting mathematical concepts.

Specific objectivesm37c0e2b077cc18f8_1528449552113_0Specific objectives

1. Identifying the circle inscribed in a triangle.

2. Constructing the circlecirclecircle inscribed in a triangletriangletriangle.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm37c0e2b077cc18f8_1528450430307_0Learning outcomes

The student:

- constructs the circle inscribed in a triangle,

- uses the properties of the circlecirclecircle inscribed in a triangletriangletriangle.

Methodsm37c0e2b077cc18f8_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm37c0e2b077cc18f8_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm37c0e2b077cc18f8_1528450127855_0Introduction

Students recall the definition of the perpendicular bisector of the line segment and its main properties.

Task

Students construct the perpendicular bisector of a line segment.

The teacher introduces the topic of the lesson: learning to inscribe a circlecirclecircle in a triangletriangletriangle.

Procedurem37c0e2b077cc18f8_1528446435040_0Procedure

Task
Open the applet and by moving the points observe the mutual position of the angle bisectors of various triangles.

[Geogebra applet]

Conclusion:

- In any triangle, the angle bisectors intersect at one point.m37c0e2b077cc18f8_1527752263647_0- In any triangle, the angle bisectors intersect at one point.

Task
Students draw a circlecirclecircle and a triangletriangletriangle in such a way that the circle touches all sides of the triangle. Then they determine the mutual position of the circlecirclecircle and obtained triangle.

[Illustration 1]

Conclusions:

- If all sides of a triangle are tangential to the circle, we say this triangle is circumscribed about a circle. The circle is said to be inscribed in a triangle.m37c0e2b077cc18f8_1527752256679_0- If all sides of a triangle are tangential to the circle, we say this triangle is circumscribed about a circle. The circle is said to be inscribed in a triangle.

We can inscribe a circle in any triangletriangletriangle. The centre of this circlecirclecircle is at the pointpointpoint of intersection of the angleangleangle bisectors of the triangle. The distance from the centre of the circle inscribed in a triangle to each side of this triangle is equal to the radius of the circle.

Task
Students draw any ABC triangle. Then they try to determine the pointpointpoint which will be equidistant from each side of the triangletriangletriangle.

Conclusion:

- The distance from the centre of the circlecentre of the circlecentre of the circle inscribed in a triangle to each side of this triangletriangletriangle is equal to the radius of the circlecirclecircle.

Task
Students draw any acute triangleacute triangleacute triangle and construct a circle inscribed in that triangle.

Task
Students draw any right‑angled triangleright‑angled triangleright‑angled triangle and construct a circlecirclecircle inscribed in that triangletriangletriangle.

Task
Students draw any obtuse triangleobtuse triangleobtuse triangle and construct a circlecirclecircle inscribed in that triangletriangletriangle.

Lesson summarym37c0e2b077cc18f8_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

- We can inscribe the circle in any triangle.
- The centre of this circle is at the point of intersection of the angle bisectors of the triangle.
- The distance from the centre of the circle inscribed in a triangle to each side of this triangle is equal to the radius of the circle.m37c0e2b077cc18f8_1527712094602_0- We can inscribe the circle in any triangle.
- The centre of this circle is at the point of intersection of the angle bisectors of the triangle.
- The distance from the centre of the circle inscribed in a triangle to each side of this triangle is equal to the radius of the circle.

Selected words and expressions used in the lesson plan

acute triangleacute triangleacute triangle

angleangleangle

angle bisectorangle bisectorangle bisector

centre of the circlecentre of the circlecentre of the circle

circlecirclecircle

obtuse triangleobtuse triangleobtuse triangle

pointpointpoint

right‑angled triangleright‑angled triangleright‑angled triangle

triangletriangletriangle