Topicmc8a52df395f1359f_1528449000663_0Topic

Triangles and their properties III

Levelmc8a52df395f1359f_1528449084556_0Level

Second

Core curriculummc8a52df395f1359f_1528449076687_0Core curriculum

XV. Symmetries. The student:

1) identifies the perpendicular bisector of a line segment and the bisector of an angle.

Timingmc8a52df395f1359f_1528449068082_0Timing

45 minutes

General objectivemc8a52df395f1359f_1528449523725_0General objective

III. Using and interpreting the representation.

Using simple, well known mathematical objects, interpreting mathematical concepts and operating on mathematical objects.

Specific objectivesmc8a52df395f1359f_1528449552113_0Specific objectives

1. Determining the bisectors of the angles of the triangle.

2. Using the properties of the medians of the triangle, determining the centroid of the trianglecentroid of the trianglecentroid of the triangle.

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

Learning outcomesmc8a52df395f1359f_1528450430307_0Learning outcomes

1. Determines the bisectors of the angles of the triangle.

2. Uses the properties of the medians of the triangle, determines the centroid of a triangle.

Methodsmc8a52df395f1359f_1528449534267_0Methods

1. Brainstorming.

2. Situational analysis.

Forms of workmc8a52df395f1359f_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmc8a52df395f1359f_1528450127855_0Introduction

The teacher introduces the topic of the lesson: revising what the bisector of an angle is and what the median of the trianglemedian of the trianglemedian of the triangle is. They will learn about the use of the median of the trianglemedian of the trianglemedian of the triangle. They will also find out how to determine the centroid of the trianglecentroid of the trianglecentroid of the triangle.

Task:
Students revise the definition of the bisector of an angle. The angle bisectorangle bisectorangle bisector is a ray that cuts an angle in half at its vertex.

[Illustration 1]

Task:
Students draw any obtuse triangle and construct the bisectors of the interior angles. They look at the mutual position of the bisectors and formulate the following conclusion:

The bisectors of the angles of the triangle intersect at one point.

Proceduremc8a52df395f1359f_1528446435040_0Procedure

Task:
Students draw the medians of the triangle and look at their mutual position. They formulate the definition of a median and write down the following conclusion.

Definition:

The median of the triangle is the line segment that joins the vertex of the triangle with the mid‑point of its opposite side. A triangle has three medians. The medians of the triangle intersect in one point, which is called the centroid of a triangle.mc8a52df395f1359f_1527752263647_0The median of the triangle is the line segment that joins the vertex of the triangle with the mid‑point of its opposite side. A triangle has three medians. The medians of the triangle intersect in one point, which is called the centroid of a triangle.

Task:
Students work individually, using computers.
They mark the mid‑points of the sides, the medians and the points of intersection of the medians. They look at the ratio at which the medians cross.

[Geogebra applet]

Having completed the task, they write down the following conclusion:

Theorem:
The centroid divides the length of each median in 2:1 ratio, starting from the vertex.

[Illustration 2]

Task:
The students solve the task using the formulated theorem.
The medians of the equilateral triangleequilateral triangleequilateral triangle intersect at the point whose distance from the vertex is 5 cm. Calculate the altitudes of this triangle.

An extra task:
In an ABC triangle the distance between the centroid and the sides are 4 cm, 6 cm and 5 cm, respectively. Calculate the sum of the distances between the centroid and the vertexes of the triangle.

Lesson summarymc8a52df395f1359f_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to remember:
- The angle bisectorangle bisectorangle bisector is a ray that cuts an angle in half at its vertex.
- The bisectors of the triangle’s angles intersect at one point.
- The median of the trianglemedian of the trianglemedian of the triangle is the line segment that joins the vertex of the trianglevertex of the trianglevertex of the triangle with the mid‑point of its opposite side. A triangle has three medians.
- The medians of the triangle intersect in one point, which is called the centroid of a triangle.
- The centroid divides the length of each median in 2:1 ratio, starting from the vertex.

Selected words and expressions used in the lesson plan

angle bisectorangle bisectorangle bisector

centroid of the trianglecentroid of the trianglecentroid of the triangle

equilateral triangleequilateral triangleequilateral triangle

isosceles triangleisosceles triangleisosceles triangle

median of the trianglemedian of the trianglemedian of the triangle

vertex of the trianglevertex of the trianglevertex of the triangle