Topicmaa161a5a8e823029_1528449000663_0Topic

The circumscribed triangletriangletriangle

Levelmaa161a5a8e823029_1528449084556_0Level

Third

Core curriculummaa161a5a8e823029_1528449076687_0Core curriculum

I. Planimetrics. The student:

10) indicates basic points of interest in triangles: the incentre of the inscribed centre of the triangletriangletriangle, the circumcentrecircumcentrecircumcentre of the triangle’s circumcircletriangle’s circumcircletriangle’s circumcircle, the centroid and uses their properties.

Timingmaa161a5a8e823029_1528449068082_0Timing

45 minutes

General objectivemaa161a5a8e823029_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesmaa161a5a8e823029_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Getting to know the Perpendicular Bisector Theorem.

3. Getting to know the structure and properties of the circumscribed triangletriangletriangle.

Learning outcomesmaa161a5a8e823029_1528450430307_0Learning outcomes

The student:

- gets to know the Perpendicular Bisector Theorem,

- gets to know the structure and properties of the circumscribed triangletriangletriangle.

Methodsmaa161a5a8e823029_1528449534267_0Methods

1. Open ear.

2. Problem discussion.

Forms of workmaa161a5a8e823029_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmaa161a5a8e823029_1528450127855_0Introduction

The student recollect their information about the bisector of a line segment and its properties using the open ear technique.

Proceduremaa161a5a8e823029_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know the structure if a circumscribed triangletriangletriangle.

Discussion – what is a circumscribed polygon? What is the property of the circumcentrecircumcentrecircumcentre of this circlecirclecircle? The students formulate hypotheses and check them making appropriate diagrams. They formulate their conclusions.

Conclusions:

- We say that a circle is a circumcircle of a polygon when it passes through all the vertices of the polygon.
- The circumcentre of a circumscribed polygon is equidistant from each of the vertices of this polygon.maa161a5a8e823029_1527752263647_0- We say that a circle is a circumcircle of a polygon when it passes through all the vertices of the polygon.
- The circumcentre of a circumscribed polygon is equidistant from each of the vertices of this polygon.

The teacher informs the students that the circumcentrecircumcentrecircumcentre of a circumcircle of a polygon is the intersection of the bisectors of the sides of this polygon.

Working in groups, the students check if there is an intersection point of the bisectors of the sides of the triangleintersection point of the bisectors of the sides of the triangleintersection point of the bisectors of the sides of the triangle.

Task for group 1
Draw an acute triangleacute triangleacute triangle. Draw the bisectors of the sides of this triangletriangletriangle. What can you notice? Write down your conclusion.

Task for group 2
Draw an obtuse triangleobtuse triangleobtuse triangle. Draw the bisectors of the sides of this triangle. What can you notice? Write down your conclusion.

Task for group 3
Draw a right triangleright triangleright triangle. Draw the bisectors of the sides of this triangletriangletriangle. What can you notice? Write down your conclusion.

Having finished, the representatives of the groups present their conclusions.

The conclusion that should be formulated:

- The bisectors of the sides of any triangletriangletriangle intersect at one point. The intersection point of the bisectors of the sides of the triangleintersection point of the bisectors of the sides of the triangleintersection point of the bisectors of the sides of the triangle is the circumcentre of the triangle circumscribed on a circlecirclecircle.

Task
The students work individually, analyzing the applet, which illustrates the method of finding the circumcentrecircumcentrecircumcentre of a triangle. They answer the question: What is the location of the circumcentre in a given type of a triangletriangletriangle? They formulate hypotheses and conclusions.

[Geogebra applet]

Conclusions:

-The circumcentre of the circle circumscribed on an acute triangle is inside this triangle.
- The circumcentre of the circle circumscribed on a right triangle is the centre of the hypotenuse.
- The circumcentre of the circle circumscribed on an obtuse triangle is outside this triangle.maa161a5a8e823029_1527752256679_0-The circumcentre of the circle circumscribed on an acute triangle is inside this triangle.
- The circumcentre of the circle circumscribed on a right triangle is the centre of the hypotenuse.
- The circumcentre of the circle circumscribed on an obtuse triangle is outside this triangle.

The students use the information to solve the tasks.

Task
Calculate the circumradius of the circlecirclecircle circumscribed on an equilateral triangletriangletriangle, whose side is 9 cm long.
Answer: $r=3\sqrt{3}$ cm.

Task
Calculate the circumradius of the circlecirclecircle circumscribed on a right triangleright triangleright triangle, whose legs are 25 cm and 15 cm long.
Answer:$r=\frac{5\sqrt{34}}{2}$ cm.

Task
One of the legs of a right triangleright triangleright triangle is 50 cm long, and the altitude led to the hypotenuse equals 40 cm. Calculate the circumradius of the circle circumscribed on this triangletriangletriangle.
Answer: $r=41\frac{2}{3}$ cm.

Task
Calculate the surface area of the circlecirclecircle circumscribed on an equilateral triangletriangletriangle, whose area equals $25\sqrt{3}$ cm.
Answer: $P=\frac{100}{3}·\mathrm{\pi }$ cmIndeks górny 2².

Having finished all the tasks, the students present their results. The teacher assesses their work and explains any doubts.

An extra task
In an isosceles triangletriangletriangle the angle at the base equals $\gamma$. Prove that że bisectors of the legs of this triangle form an angle which measures $2·\gamma$.

Lesson summarymaa161a5a8e823029_1528450119332_0Lesson summary

The students do the consolidation tasks. Then, they summarize the class, formulate the conclusions to be remembered.

- We say that a circle is a circumcircle of a polygon when it passes through all the vertices of the polygon.
- The circumcentre of a circumscribed polygon is equidistant from each of the vertices of this polygon.
- The bisectors of the sides of any triangle intersect at one point, which is the circumcentre of the circle circumscribed on this triangle.
- The circumcentre of the circle circumscribed on an acute triangle is inside this triangle.
- The circumcentre of the circle circumscribed on a right triangle is the centre of the hypotenuse.maa161a5a8e823029_1527712094602_0- We say that a circle is a circumcircle of a polygon when it passes through all the vertices of the polygon.
- The circumcentre of a circumscribed polygon is equidistant from each of the vertices of this polygon.
- The bisectors of the sides of any triangle intersect at one point, which is the circumcentre of the circle circumscribed on this triangle.
- The circumcentre of the circle circumscribed on an acute triangle is inside this triangle.
- The circumcentre of the circle circumscribed on a right triangle is the centre of the hypotenuse.

Selected words and expressions used in the lesson plan

acute triangleacute triangleacute triangle

bisector of the triangle’s sidebisector of the triangle’s sidebisector of the triangle’s side

circlecirclecircle

circumcentrecircumcentrecircumcentre

circumradius of the triangle’s circumcirclecircumradius of the triangle’s circumcirclecircumradius of the triangle’s circumcircle

intersection point of the bisectors of the sides of the triangleintersection point of the bisectors of the sides of the triangleintersection point of the bisectors of the sides of the triangle

obtuse triangleobtuse triangleobtuse triangle

right triangleright triangleright triangle

triangletriangletriangle

triangle’s circumcircletriangle’s circumcircletriangle’s circumcircle