Topicm6c5e73796a2377f0_1528449000663_0Topic

A circle circumscribed about a triangletriangletriangle

Levelm6c5e73796a2377f0_1528449084556_0Level

Second

Core curriculumm6c5e73796a2377f0_1528449076687_0Core curriculum

XV. Symmetries. The student:

1) identifies the perpendicular bisector and the angle bisector of line segments;

2) knows and uses practically the basic properties of the perpendicular bisector and the angle bisector of the line segmentline segmentline segment the sample exercise below: 
The vertex C of the ABCD rhombus is located on the perpendicular bisectors of the sides AB and AD. Calculate the angles of this rhombus.

Timingm6c5e73796a2377f0_1528449068082_0Timing

45 minutes

General objectivem6c5e73796a2377f0_1528449523725_0General objective

Using simple, well known mathematical objects, interpreting mathematical concepts.

Specific objectivesm6c5e73796a2377f0_1528449552113_0Specific objectives

1. Identifying a circle circumscribed about a triangletriangletriangle.

2. Constructing a circle circumscribed about a triangletriangletriangle.

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

Learning outcomesm6c5e73796a2377f0_1528450430307_0Learning outcomes

The student:

- constructs a circle circumscribed about a triangletriangletriangle,

- uses the properties of a circle circumscribed about a triangletriangletriangle.

Methodsm6c5e73796a2377f0_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm6c5e73796a2377f0_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm6c5e73796a2377f0_1528450127855_0Introduction

Students recall the definition of the perpendicular bisector of the line segmentline segmentline segment and its main properties.

Task

Students construct the perpendicular bisector of a line segmentline segmentline segment.

The teacher introduces the topic of the lesson: learning to circumscribe a circle on any triangletriangletriangle.

Procedurem6c5e73796a2377f0_1528446435040_0Procedure

Task

Students work individually using computers.

Their task is to observe the mutual position of the perpendicular bisectors of the sides of various triangles.

[Geogebra applet 1]

The conclusions students should make.

- In any triangle, the perpendicular bisectors of the side intersect at one point.
- If the triangle is acute, the point of intersection is inside the triangle.
- If the triangle is right‑angled, the point of intersection is in the middle of the hypotenuse.
- If the triangle is obtuse, the point of intersection is outside of the triangle.m6c5e73796a2377f0_1527752263647_0- In any triangle, the perpendicular bisectors of the side intersect at one point.
- If the triangle is acute, the point of intersection is inside the triangle.
- If the triangle is right‑angled, the point of intersection is in the middle of the hypotenuse.
- If the triangle is obtuse, the point of intersection is outside of the triangle.

Task

Students draw a circle and then connect any three points on the circumference with the line segmentline segmentline segment. Together they identify the mutual location of the circle and the obtained triangletriangletriangle.

[Illustration 1]

The conclusion students should make.

- If all vertices of the triangle are located on the circle, this circle is said to be circumscribed about the triangle. The triangle is inscribed in the circle.m6c5e73796a2377f0_1527752256679_0- If all vertices of the triangle are located on the circle, this circle is said to be circumscribed about the triangle. The triangle is inscribed in the circle.

Task

Students work individually, using computers.

Their task is to observe the construction of a circle circumscribed about any triangletriangletriangle.

[Geogebra applet 2]

Task

Students draw any acute triangletriangletriangle and construct a circle circumscribed about that triangle.

Task

Students draw any right‑angled triangleright‑angled triangleright‑angled triangle and construct a circle circumscribed about that triangle.

Task

Students draw any obtuse triangletriangletriangle and construct a circle circumscribed about that triangle.

The conclusions students should make.

- The centre of the circle circumscribed about an acute triangle is located inside the triangletriangletriangle.

- The centre of the circle circumscribed about a right‑angled triangleright‑angled triangleright‑angled triangle is located in the middle of the hypotenuse.

- The centre of the circle circumscribed about an obtuse triangle is located outside of the triangle.

Lesson summarym6c5e73796a2377f0_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

- In any triangle, the perpendicular bisectors of the side intersect at one point.
- If the triangle is acute, the point of intersection is inside the triangle.
- If the triangle is right‑angled, the point of intersection is in the middle of the hypotenuse.
- If the triangle is obtuse, the point of intersection is outside the triangle.m6c5e73796a2377f0_1527752263647_0- In any triangle, the perpendicular bisectors of the side intersect at one point.
- If the triangle is acute, the point of intersection is inside the triangle.
- If the triangle is right‑angled, the point of intersection is in the middle of the hypotenuse.
- If the triangle is obtuse, the point of intersection is outside the triangle.

- We can circumscribe a circle about any triangletriangletriangle.

- The centre of the circle circumscribed about an acute triangle is located inside the triangletriangletriangle.

- The centre of the circle circumscribed about a right‑angled triangleright‑angled triangleright‑angled triangle is located in the middle of the hypotenuse.

- The centre of the circle circumscribed about an obtuse triangletriangletriangle is located outside of the triangle.

Selected words and expressions used in the lesson plan

circlecirclecircle

line segmentline segmentline segment

pointpointpoint

right‑angled triangleright‑angled triangleright‑angled triangle

sidesideside

triangletriangletriangle