# Three Days in the Lab

### by Margaret Sinclair

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Preparation

The following constructions and investigations require that students have had a brief introduction to the tools in the Geometer's Sketchpad, and a review of some key terms in triangle and circle geometry.

I have found it helpful to have students work in pairs.

Evaluation

As students complete each question I check their work and probe their understanding by questioning them about the construction. Their in-class mark is counted as 25% of the mark for the unit. For other ideas consult Tips for Successful Lab Sessions.

Day One Assignment

1. Construct the centroid of a triangle.

2. Construct the incenter of a triangle and the incircle.

3. Construct the circumcenter of a triangle and the circumcircle.

4. Construct the orthocenter of a triangle.

5. Three of the four points of concurrency (centroid, incenter, circumcenter and orthocenter) are always collinear. The line that contains them is called the Euler Line. Discover which three are collinear and demonstrate using Sketchpad, giving an explanation of your findings.

6. Draw a triangle and construct the following nine points:

1. the midpoints of the sides;
2. the intersection of altitudes with the sides;
3. the midpoints of segments from orthocenter to the vertices.

Construct a circle through any three of these nine points. What do you notice? How is this circle related to the Euler line that you drew in question 5?

7. Explain how you can find:

1. the point that is equidistant from the sides of a triangle;
2. The point that is equidistant from the vertices of a triangle.

Day Two Assignment

1. Demonstrate using Sketchpad that the measure of the side of an equilateral triangle circumscribed about a circle is two times the measure of the side of the inscribed equilateral triangle.

2. Carry out the following investigation:

1. Construct any triangle.
2. Construct an equilateral triangle on each side of your triangle.
3. Construct the centroids of these equilateral triangles.

What conclusions can you draw about the triangle formed by joining the centroids?

3. Draw a pentagon and find:

1. the sum of the interior angles;
2. the sum of the exterior angles.

4. Repeat question 3 using a hexagon.

5. From your findings in questions 2 and 3 conjecture the following:

1. the sum of the interior angles of a 20-gon
2. the sum of the exterior angles of a 20-gon

Day Three Assignment

1. Draw a circle and demonstrate:

1. the relationship between inscribed angles subtended by the same arc
2. the relationship between an inscribed angle and a central angle subtended by the same arc
3. the measure of an angle subtended by a semi-circle

2. Construct a quadrilateral inscribed in a circle and investigate the relationship between opposite angles in the quadrilateral.

3. Demonstrate the following, if possible, or explain why it is not possible.
A circle can be drawn through:

1. the 4 vertices of any parallelogram;
2. the 4 vertices of any rectangle;
3. the 4 vertices of any rhombus;
4. the 5 vertices of any regular pentagon;
5. the n vertices of any regular n-gon.

4. Construct 5 points on a circle. Join the points to form a star. Find the sum of the angles in the star.

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