Okrąg opisany na trójkącie

Learning objectives

• You will develop the following competences: communicating in English, mathematical, IT and basic scientific and technical competence, your learning skills.

• You will get to know the Perpendicular Bisector Theorem.

• You will get to know the structure and properties of the circumscribed triangle.

Learning effect

• You will learn the Perpendicular Bisector Theorem.

• You will learn the structure and properties of the circumscribed triangle.

Recollect what you know about the bisector of a line segment and its properties. The aim of the class is getting to know the structure if a circumscribed triangletriangletriangle.

Think - what is a circumscribed polygon? What is the property of the circumcentrecircumcentrecircumcentre of this circlecirclecircle? Check your hypotheses making appropriate diagrams. Formulate your conclusions.

• We say that a circlecirclecircle is a circumcircle of a polygon when it passes through all the vertices of the polygon.

• The circumcentrecircumcentrecircumcentre of a circumscribed polygon is equidistant from each of the vertices of this polygon.

Find in the available sources what is another name of the point which is the centre of a polygon’s circumcircle.

The answer should be:

• The circumcentrecircumcentrecircumcentre of a circumcircle of a polygon is the intersection of the bisectors of the sides of this polygon.

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.

Conclusion:

• 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 circumcentrecircumcentrecircumcentre of the triangle circumscribed on a circlecirclecircle.

Task 2

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nagranie abstraktu

nagranie abstraktu

Nagranie dostępne na portalu epodreczniki.pl

nagranie abstraktu

Analyse the applet, which illustrates the method of finding the circumcentre of a triangletriangletriangle. Answer the question: What is the location of the circumcentrecircumcentrecircumcentre in a given type of a triangle? Formulate your conclusions.

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Aplikacja geogebra: Okrąg opisany na trójkącie. Poniżej znajduje się galeria będąca wersją alternatywną dla aplikacji.

Aplikacja geogebra: Okrąg opisany na trójkącie. Poniżej znajduje się galeria będąca wersją alternatywną dla aplikacji.

Zasób interaktywny dostępny pod adresem https://zpe.gov.pl/a/DFypmS2U

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Rysunek trójkąta ABC z zaznaczonymi liniami przerywanymi środkowymi boków AB i BC. Punkt ich przecięcia oznaczono S. Zaznaczono odcinki łączące punkt S z wierzchołkami trójkąta.

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Rysunek trójkąta ABC z zaznaczonymi liniami przerywanymi środkowymi wszystkich boków. Punkt ich przecięcia oznaczono S. Zaznaczono odcinki łączące punkt S z wierzchołkami trójkąta. Okrąg o środku w punkcie S przechodzący przez wierzchołki trójkąta.

• The circumcentrecircumcentrecircumcentre of the circle circumscribed on an acute triangleacute triangleacute triangle is inside this triangletriangletriangle.

• The circumcentre of the circlecirclecircle circumscribed on a right triangleright triangleright triangle is the centre of the hypotenuse.

• The circumcentre of the circle circumscribed on an obtuse triangleobtuse triangleobtuse triangle is outside this triangle.

Use the information to solve the tasks.

Having finished all the tasks, do the consolidation tasks. Then, formulate the conclusions that you need to remember.

• We say that a circlecirclecircle is a circumcircle of a polygon when it passes through all the vertices of the polygon.

• The circumcentrecircumcentrecircumcentre of a circumscribed polygon is equidistant from each of the vertices of this polygon.

• The bisectors of the sides of any triangletriangletriangle intersect at one point, which is the circumcentre of the circlecirclecircle circumscribed on this triangle.

• The circumcentre of the circle circumscribed on an acute triangleacute triangleacute triangle is inside this triangletriangletriangle.

• The circumcentrecircumcentrecircumcentre of the circle circumscribed on a right triangleright triangleright triangle is the centre of the hypotenuse.

• The circumcentre of the circlecirclecircle circumscribed on an obtuse triangleobtuse triangleobtuse triangle is outside this triangletriangletriangle.

Exercises

Glossary

Keywords

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

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

triangletriangletriangle

triangle’s circumcircletriangle’s circumcircletriangle’s circumcircle