Topicm2c454dbee7f1ddba_1528449000663_0Topic

Circle inscribed in the right triangle

Levelm2c454dbee7f1ddba_1528449084556_0Level

Third

Core curriculumm2c454dbee7f1ddba_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

1) finds the radii and the diameters of circles, the lengths of arcs and tangent segments, also with the use of the Pythagorean theorem.

Timingm2c454dbee7f1ddba_1528449068082_0Timing

45 minutes

General objectivem2c454dbee7f1ddba_1528449523725_0General objective

Using and interpreting the representation. Using mathematical objects and manipulating them, interpreting mathematical concepts. Selecting and creating mathematical models to solve practical and theoretical problems.

Specific objectivesm2c454dbee7f1ddba_1528449552113_0Specific objectives

1. Formulating the theorem about the radiusradiusradius of a circle inscribed in a right triangle.

2. Applying properties of a circle inscribed in a right triangle to solve problems.

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

Learning outcomesm2c454dbee7f1ddba_1528450430307_0Learning outcomes

The student:

- formulates the theorem about the radius of a circle inscribed in a right triangle,

- applies the properties of a circle inscribed in a right triangle to solve problems.

Methodsm2c454dbee7f1ddba_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm2c454dbee7f1ddba_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm2c454dbee7f1ddba_1528450127855_0Introduction

The aim of the introductory part is to remind the students of mathematical concepts that will be used in the lesson: the Pythagorean theorem, the theorem about the tangents to a circle, the construction of a circle inscribed in a triangle.

The teacher divides the students into small groups. The task of each group will be to collect and systematize information related to one of the above concepts. The groups present and systematize the collected information.

Procedurem2c454dbee7f1ddba_1528446435040_0Procedure

Students work individually or in pairs, using computers. They observe how the length of the radiusradiusradius of an inscribed circle changes.

Task
Open the Geogebra applet: „Circle inscribed in a right triangle”. Change the position of point A. Observe how, depending on the length of the radius, the lengths of the other color‑marked segments change. Answer the following questions:

1. Is the triangle PAR right? Why?.
2. Is the quadrilateral SEAF square? Why?
3. Is it possible to determine the length of the hypotenusehypotenusehypotenuse depending on the length of the other sides of triangle and the radius of the circle? How?

Conclusion:

- There is a relationship between the radiusradiusradius of a circle inscribed in a right triangle and the length of its sides.

The students, under the teacher’s supervision, derive teh formula for the radius of a circle inscribed in the right trianglecircle inscribed in the right trianglecircle inscribed in the right triangle.

[Illustration 1]

This phase of lesson should be summarised with the formulation of the following theorem.

Theorem – the radius of a circle inscribed in the right triangle.

- The radius r of a circle inscribed in the right trianglecircle inscribed in the right trianglecircle inscribed in the right triangle with the hypotenusehypotenusehypotenuse c and the other sides a and b equals:

Students work individually and then discuss the results.

Task
Find the radius of a circle inscribed in the right triangle whose legs are 9 cm and 12 cm.m2c454dbee7f1ddba_1527752256679_0Find the radius of a circle inscribed in the right triangle whose legs are 9 cm and 12 cm.

Task
The radius of a circle inscribed in the right triangle is 2 cm. The point of tangency divides the hypotenuse in a ratio of 2:3. Calculate the perimeter of this triangle.m2c454dbee7f1ddba_1527752263647_0The radius of a circle inscribed in the right triangle is 2 cm. The point of tangency divides the hypotenuse in a ratio of 2:3. Calculate the perimeter of this triangle.

Task
The square ABCD with side a is divided diagonally to two triangles ABC and CDA. A circle is inscribed in the triangle ABC. Find the radiusradiusradius of this circle.

An extra task:
The area of the triangles with sides a, b, c is equal to:

$P=p\cdot r$

where:
$p=\frac{a+b+c}{2}$
$r$ – the radius of a circle inscribed in this triangle.
Use this relationship and find the formula for the radius of a circle inscribed in the right triangle.

Lesson summarym2c454dbee7f1ddba_1528450119332_0Lesson summary

Students do the revision exercises.

Then, together they summarise the class, by formulating the following theorem.

- The radius r of a circle inscribed in the right trianglecircle inscribed in the right trianglecircle inscribed in the right triangle with the hypotenusehypotenusehypotenuse c and the other sides a and b equals:

Selected words and expressions used in the lesson plan

area of trianglearea of trianglearea of triangle

circle inscribed in the right trianglecircle inscribed in the right trianglecircle inscribed in the right triangle

hypotenusehypotenusehypotenuse

point of tangencypoint of tangencypoint of tangency

radiusradiusradius

short multiplication formulashort multiplication formulashort multiplication formula

tangent to the circletangent to the circletangent to the circle