Topicm03a7c364ef4d0cdc_1528449000663_0Topic

Bisectors of angles in the triangletriangletriangle

Levelm03a7c364ef4d0cdc_1528449084556_0Level

Third

Core curriculumm03a7c364ef4d0cdc_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

7) applies the following theorems: the Thales’s theorem, the inverse Thales’s theorem, the angle bisectorangle bisectorangle bisector theorem and the theorem about the angleangleangle between the tangent and the chord;

10) identifies specific basic points in triangles: the incenterincenterincenter, the center of the circumscribed circle, the orthocenter, the center of gravity and applies their properties.

Timingm03a7c364ef4d0cdc_1528449068082_0Timing

45 minutes

General objectivem03a7c364ef4d0cdc_1528449523725_0General objective

Using and interpreting the representation. Using mathematical objects and manipulating them, interpreting mathematical concepts.

Noticing regularities, similarities and analogies and formulating relevant conclusions.

Specific objectivesm03a7c364ef4d0cdc_1528449552113_0Specific objectives

1. Systematization of knowledge about the properties of angle bisectorangle bisectorangle bisector.

2. Applying the angle bisector theorem to solve problem.

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

Learning outcomesm03a7c364ef4d0cdc_1528450430307_0Learning outcomes

The student:

- applies the properties of the angle bisector to solve problems,

- applies the angle bisectorangle bisectorangle bisector theorem.

Methodsm03a7c364ef4d0cdc_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm03a7c364ef4d0cdc_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm03a7c364ef4d0cdc_1528450127855_0Introduction

The teacher introduces the topic of the lesson: the bisectors of an angleangleangle and their properties in triangles.

Students recall the definition of the angle bisectorangle bisectorangle bisector.

Definition – the angle bisector.

- The angle bisector is a ray that begins at the vertex of the angle and divides the angle into two congruent angles.m03a7c364ef4d0cdc_1527752263647_0- The angle bisector is a ray that begins at the vertex of the angle and divides the angle into two congruent angles.

Students work in pairs and recall the consecutive steps of the construction of the angle bisector. They describe the main properties of the angle bisectorangle bisectorangle bisector.

Task
Draw any acute angle and construct the bisector of this angleangleangle.

Theorem – the pointpointpoint on the bisector of an angle.

- If a point lies on the bisector of an angle, then it is equidistant from the side of the angle.
In other words:
The bisector of an angle is a set of points whose distance from the sides of the angle is the same. Each point of the angle bisector is equidistant from the sides of the angle.m03a7c364ef4d0cdc_1527752256679_0- If a point lies on the bisector of an angle, then it is equidistant from the side of the angle.
In other words:
The bisector of an angle is a set of points whose distance from the sides of the angle is the same. Each point of the angle bisector is equidistant from the sides of the angle.

[Illustration 1]

Procedurem03a7c364ef4d0cdc_1528446435040_0Procedure

Students work individually or in pairs, using computers. They identify properties of the pointpointpoint where the angle bisectors of the triangletriangletriangle intersect.

Task
Open the applet. Change the position of vertexvertexvertex of the triangle and observe the intersection pointpointpoint of the angle bisectors of the triangletriangletriangle.

[Geogebra applet]

Decide whether:
1. the angle bisectors of the triangle intersect at one point,
2. the point of intersection is inside the triangle,
3. the pointpointpoint of intersection is equidistant from the sides of the triangletriangletriangle.

Conclusion:

- The angle bisectors of the triangletriangletriangle intersect at one point. This pointpointpoint is the center of the incircleincircleincircle, also called the incenterincenterincenter of the triangle.

Students work in pairs and discover the main property of angle bisectors in the triangle.

Task
In the ABC triangletriangletriangle, the bisector of the ABC angleangleangle has been drawn. This angle bisectorangle bisectorangle bisector intersects the AB sidesideside of the triangle at pointpointpoint D. Find $\frac{\left|\begin{array}{c}AD\end{array}\right|}{\left|\begin{array}{c}DB\end{array}\right|}$.

A hint:
Note that the ADC and BDC triangles share the altitude from the vertexvertexvertex C. The altitudes of these triangles from vertex D are equal (cf. the property of the angle bisectorangle bisectorangle bisector, i.e. pointpointpoint D is equidistant from the sides of these triangles). Using the formula for the areaareaarea of the triangletriangletriangle, show that $\frac{\left|\begin{array}{c}AD\end{array}\right|}{\left|\begin{array}{c}DB\end{array}\right|}=\frac{{\displaystyle \left|\begin{array}{c}AC\end{array}\right|}}{{\displaystyle \left|\begin{array}{c}BC\end{array}\right|}}$.

Students summarize their work by formulating the following theorem.

Theorem – the angle bisectorangle bisectorangle bisector theorem.

- The angle bisector of the triangle divides the opposite side into segments that are proportional to the adjacent sides. m03a7c364ef4d0cdc_1527712094602_0- The angle bisector of the triangle divides the opposite side into segments that are proportional to the adjacent sides. 

Students work individually and solve problems applying the theorems they learned and then discuss the results.

Task
- In an isosceles triangle, the bisectors of the base angles intersect at 150°. Find the measures of all angles in this triangletriangletriangle.
- The triangle with the sides 5, 7 and 10 is given. The bisector of one angleangleangle has divided the longest side into two segments. Find the lengths of these segments.
- The angle bisectorangle bisectorangle bisector divides the opposite sidesideside of the triangletriangletriangle into segments of the following lengths: 5 and 6. The perimeterperimeterperimeter of the triangle is 33. Find the lengths of the other two sides.

An extra task:
The angle bisectorangle bisectorangle bisector divides the opposite sidesideside of the triangle into segments of the following lengths: 3 and 6. Derive the formula for the perimeter of the triangletriangletriangle knowing the length of one of its sides. Is there a triangle that meets this condition which has the perimeterperimeterperimeter equal to 16 cm?

A hint:
Remember that the inequality of the triangletriangletriangle should be satisfied.

Lesson summarym03a7c364ef4d0cdc_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they summarize the class, by formulating the main definitions and conclusions to remember.

- The angle bisectorangle bisectorangle bisector divides an angle into two congruent angles.

- The bisector of an angle is a set of points whose distance from the sides of the angle is the same.

- The angle bisectors of the triangle intersect at one point. This pointpointpoint is the center of the incircleincircleincircle, also called the incenterincenterincenter of the triangle.

- The angle bisectorangle bisectorangle bisector of the triangle divides the opposite sidesideside into segments that are proportional to the adjacent sides.

Selected words and expressions used in the lesson plan

angleangleangle

angle bisectorangle bisectorangle bisector

areaareaarea

incenterincenterincenter

incircleincircleincircle

perimeterperimeterperimeter

pointpointpoint

sidesideside

triangletriangletriangle

vertexvertexvertex