Topicm46a3c3da9afe995d_1528449000663_0Topic

Particular types of right‑angled trianglesparticular types of right‑angled trianglesParticular types of right‑angled triangles

Levelm46a3c3da9afe995d_1528449084556_0Level

Second

Core curriculumm46a3c3da9afe995d_1528449076687_0Core curriculum

VIII. Properties of planar geometric figures. The student:

8) knows and uses the Pythagorean theoremPythagorean theoremPythagorean theorem (without the converse theorem) in practical situations.

Timingm46a3c3da9afe995d_1528449068082_0Timing

45 minutes

General objectivem46a3c3da9afe995d_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm46a3c3da9afe995d_1528449552113_0Specific objectives

1. Identifying the properties of right‑angled and isosceles triangles.

2. Identifying the properties of right‑angled triangles whose acute angles are 30° and 60°.

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

Learning outcomesm46a3c3da9afe995d_1528450430307_0Learning outcomes

The student:

- identifies the properties of right‑angled and isosceles triangles,

- identifies the properties of right‑angled triangles whose acute angles are 30° and 60°.

Methodsm46a3c3da9afe995d_1528449534267_0Methods

1. Discussion.

2. Brainstorming.

Forms of workm46a3c3da9afe995d_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm46a3c3da9afe995d_1528450127855_0Introduction

The teacher introduces the subject of the lesson: identifying the properties of right‑angled and isosceles triangles whose acute angled are 30° and 60°.

Task
Students revise the Pythagorean TheoremPythagorean theoremPythagorean Theorem and the theorem about the sum of angles in the triangle.

Procedurem46a3c3da9afe995d_1528446435040_0Procedure

Task
Students observe how the length of the diagonal of the squaresquaresquare changes depending on the length of the side. Determine the formula for the diagonal of the squarediagonal of the squarediagonal of the square whose side is $a$ and the analogical relation for a right‑angled and isosceles triangle.

[Geogebra applet]

Conclusion:

The diagonal of the square whose side is $a$ is equal to $a\sqrt{2}$.
In the right‑angled triangle whose cathetuses are equal to $a$, the hypotenuse is equal to $a\sqrt{2}$.m46a3c3da9afe995d_1527752263647_0The diagonal of the square whose side is $a$ is equal to $a\sqrt{2}$.
In the right‑angled triangle whose cathetuses are equal to $a$, the hypotenuse is equal to $a\sqrt{2}$.

[Illustraton 1]

Students use the relevant formulas in the exercises:

Task
Calculate the perimeter of the right‑angled and isosceles triangle whose hypotenuses are 4 cm.

Task
Calculate the perimeter of the isosceles trapezium whose bases are 6 cm and 10 cm and whose angle between the leg and the longer base is 45°.

Task
The obtuse angleobtuse angleobtuse angle of a parallelogram is 135°. The longer side is 12 cm. Calculate the perimeter of the parallelogram, knowing that its altitude is 4 cm.

Task
Together students think about the relations between the sides in the right‑angled triangleright‑angled triangleright‑angled triangle whose acute angles are 30° and 60°.

They write down the conclusion.

Conclusion:

In the right‑angled triangle whose acute angles are 30°, 60° and whose hypotenuse is $2a$:
- The cathetuse located opposite to the angle 30° is equal to $a$.
- The cathetuse located opposite to the angle 60° is equal to $a\sqrt{3}$.m46a3c3da9afe995d_1527752256679_0In the right‑angled triangle whose acute angles are 30°, 60° and whose hypotenuse is $2a$:
- The cathetuse located opposite to the angle 30° is equal to $a$.
- The cathetuse located opposite to the angle 60° is equal to $a\sqrt{3}$.

[Illustraton 2]

Students use the defined properties in exercises.

Task
Calculate the perimeter of the triangle presented in the picture:

[Illustraton 3]

An extra task
The relation between the acute angles in a right‑angled triangleright‑angled triangleright‑angled triangle is 2 : 1. The shorter cathetusecathetusecathetuse is equal to $3\sqrt{3}$. Calculate the perimeter of this triangle.

Lesson summarym46a3c3da9afe995d_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

Conclusion:

The relation between the sides in a right‑angled and an isosceles triangle.

[Illustraton 1]

Conclusion:

The relations between the sides in the right‑angled triangleright‑angled triangleright‑angled triangle whose acute angles are 30° and 60°:

[Illustraton 2]

Selected words and expressions used in the lesson plan

acute angleacute angleacute angle

cathetusecathetusecathetuse

diagonal of the squarediagonal of the squarediagonal of the square

hypotenusehypotenusehypotenuse

measure of the anglemeasure of the anglemeasure of the angle

obtuse angleobtuse angleobtuse angle

particular types of right‑angled trianglesparticular types of right‑angled trianglesparticular types of right‑angled triangles

Pythagorean theoremPythagorean theoremPythagorean theorem

right‑angled triangleright‑angled triangleright‑angled triangle

squaresquaresquare