Topicm1c606a0b4e8b854c_1528449000663_0Topic

The altitudealtitudealtitude of a triangletriangletriangle

Levelm1c606a0b4e8b854c_1528449084556_0Level

Second

Core curriculumm1c606a0b4e8b854c_1528449076687_0Core curriculum

XI. Calculations in geometry. The student:

2) calculates the area of: the triangletriangletriangle, the square, the rectangle, the rhombus, the parallelogram and the trapezium presented in the drawing and in practical situations, including data requiring a conversion of units and in situations when the dimensions are not typical, for example the area of the triangle with a sidesideside of 1 km and the altitudealtitudealtitude of 1 mm.

Timingm1c606a0b4e8b854c_1528449068082_0Timing

45 minutes

General objectivem1c606a0b4e8b854c_1528449523725_0General objective

Creating and interpreting mathematical texts, presenting data graphically.

Specific objectivesm1c606a0b4e8b854c_1528449552113_0Specific objectives

1.Recognising and constructing the altitude of the triangletriangletriangle.

2. Describing the properties of the altitudealtitudealtitude of the triangle.

3. Communicating in English; developing mathematical and basic scientific, technical and digital competences; developing learning skills.

Learning outcomesm1c606a0b4e8b854c_1528450430307_0Learning outcomes

The student:

- recognises the altitudealtitudealtitude of acute, right and obtuse triangles,

- marks the altitude of the triangle using the set squareset squareset square or making an appropriate construction,

- describes the basic properties of the altitudealtitudealtitude of the triangletriangletriangle,

- describes in English the altitude and its properties.

Methodsm1c606a0b4e8b854c_1528449534267_0Methods

1. Practical exercises.

2. Situational analysis.

Forms of workm1c606a0b4e8b854c_1528449514617_0Forms of work

1. Individual work.

2. Class work.

Lesson stages

Introductionm1c606a0b4e8b854c_1528450127855_0Introduction

Before the lesson students cut out in self‑adhesive paper three triangles (the acute‑angled, the right‑angled and the obtuse‑angled).

The teacher introduces the topic of the lesson: learning about the altitude of the triangletriangletriangle. They are going to find out the method for recognising the altitudealtitudealtitude and constructing it by using the set squareset squareset square.

Procedurem1c606a0b4e8b854c_1528446435040_0Procedure

The teacher draws the acute triangletriangletriangle and asks the following questions:

How many segments connecting the triangle vertexvertexvertex with its opposite sidesideside can be drawn?

Which of these segments is the shortest?

The students should draw the following conclusion:

- The number of these segments is infinite but the shortest is the one that is perpendicularperpendicularperpendicular to the opposite sidesideside.

The teacher introduces the notion of the altitudealtitudealtitude of the triangletriangletriangle, its definition accompanied by a sample figurefigurefigure of altitude of the acute triangle:

The altitude of the triangle is the segment connecting its vertex with the opposite side or its extension which is perpendicular to this side.
The triangle has got three altitudes.
The altitude of the triangle is marked with the small letter h.m1c606a0b4e8b854c_1527752263647_0The altitude of the triangle is the segment connecting its vertex with the opposite side or its extension which is perpendicular to this side.
The triangle has got three altitudes.
The altitude of the triangle is marked with the small letter h.

The teacher asks the following questions:

Is it possible to draw a few altitudes out of one vertex?

How many altitudes does the triangletriangletriangle have?

The students should draw the following conclusions:

- one altitudealtitudealtitude can be drawn out of one vertexvertexvertex,

- the triangle has three altitudes.

Task

The students work individually using the computers. Their task is to observe how the mutual position of the altitudes of the triangle change depending on the changes of the positions of its vertices.

[Geogebra applet]

Task

After finishing the task the students present the observation results by answering the following questions:

Is it possible for the altitude of the triangletriangletriangle to be its sidesideside at the same time?

Is it possible for the altitudealtitudealtitude of the triangle to be situated outside of the figurefigurefigure?

Do the altitudes of the triangle intersect at one point? If they do so, what type of triangle is it?

Do the straight lines containing the altitudes of the triangle always intersect at one point?

The students should draw the following conclusions:

- in the right‑angled triangle two altitudes are its sides at the same time,

- in the obtuse‑angled triangleobtuse‑angled triangleobtuse‑angled triangle two altitudes are situated outside of the triangletriangletriangle,

- the altitudes intersect at one point in both acute‑angled and right‑angled triangles,

- the straight lines including the triangle altitudes intersect at one point in every triangle.

The teacher asks the following question:

How can you check if the segmentsegmentsegment you have drawn is the altitudealtitudealtitude of the triangletriangletriangle?

The students should draw the following conclusion:

- The altitudealtitudealtitude is the shortest segment connecting the vertexvertexvertex with its opposite sidesideside (or its extension) which is called the base. This segmentsegmentsegment must be perpendicularperpendicularperpendicular to the base; you can check it using a set squareset squareset square.

The teacher explains how to construct the altitude of the triangle.

You can draw the altitude of the triangle using the perpendicular arms of the set square. One arm should cover the base of the triangle and the other one should intersect the opposite vertex.
In obtuse‑angled triangles the altitudes drawn out of the vertices of the obtuse angles are situated outside of the triangle. Before we draw the altitudes we should extend these sides.m1c606a0b4e8b854c_1527752256679_0You can draw the altitude of the triangle using the perpendicular arms of the set square. One arm should cover the base of the triangle and the other one should intersect the opposite vertex.
In obtuse‑angled triangles the altitudes drawn out of the vertices of the obtuse angles are situated outside of the triangle. Before we draw the altitudes we should extend these sides.

Look at the figurefigurefigure below. Think of the place where you should put your set square to draw the altitude.

[Illustration 1]

Task

Students stick the triangles they have prepared into their notebooks. Using the set squareset squareset square they draw all altitudes of these triangles.

An extra Task

Construct the altitudealtitudealtitude of an acute triangletriangletriangle using the compass. 

Lesson summarym1c606a0b4e8b854c_1528450119332_0Lesson summary

The students complete the summary exercises.

Then they summarise the lesson, drawing conclusions to be memorised:

- The altitudealtitudealtitude of the triangle is the shortest segmentsegmentsegment connecting its vertexvertexvertex with the opposite side (or the extended sidesideside) at the right angle.

- We can draw three altitudes in each triangle.

- The straight lines containing the altitudes of the triangle always intersect at one point regardless of the type of the triangle.

Selected words and expressions used in the lesson plan

altitudealtitudealtitude

figurefigurefigure

obtuse‑angled triangleobtuse‑angled triangleobtuse‑angled triangle

perpendicularperpendicularperpendicular

segmentsegmentsegment

set squareset squareset square

sidesideside

triangletriangletriangle

vertexvertexvertex