# Constructive Triangles: Related Activities

## Contents

6-9

### Presentation

1. Construct the first star. Notice that the triangles meet at a point in the center. We must divide this star into two equal parts, leaving six triangles on one side and six on the other. Many possibilities exist; simply choose on and slide the triangles away to leave a gap.
We want to make the point at the top of one side meet the point on the top of the other. Slide one half along and then towards the other to make the two points meet at the top. We see that they have met at the bottom also, and where there was a point in the center there is now a line segment.

Again divide the figure in half, this time along the other side of the triangle which was displaced before. Separate the two halves to leave a gap. Identify the two points at the top and bottom which should meet. Slide one half into position. We see that a quadrilateral (a rhombus) has been created at the center.

Continue in the same manner, identifying the figure formed at the center each time: equilateral hexagon, equilateral octagon, equilateral decagon, equilateral and equiangular, therefore regular dodecagon. This is the first diaphragm. It is like the diaphragm of a camera. Bring one in to demonstrate.

2. Construct the second star. As before, divide into two equal parts. Slide one side so that the vertices of the extreme angles meet. Note the change from a point to a line segment. Continue naming each of the figures made, ending with the equilateral and equiangular, therefore - regular hexagon.

3. Construct the third star. Divide as before and slide one half. In this only the point, line segment and square are formed in the center. This is the third diaphragm.

Note: This third diaphragm will serve as a point of reference for two algebraic demonstrations of the Pythagorean theorem.

4. The children draw, cut, and paste the stars and diaphragms.

5. Older children may solve for the areas of the diaphragms and their internal figures, and find the relationship between them.

6. Constructing the second or third star, the child forms other figures by fitting in the angles.

7. Encourage further explorations using these triangles.

### Purpose

Direct Aim:

•  Exploration of shapes using triangles.

Indirect Aim:

•  Preparation for the sum of exterior and interior angles.