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Of All the Areas

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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LOGO Challenge 5 - Patch

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

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LOGO Challenge 6 - Triangles and Stars

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

Can You Explain Why?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Kkytha (Wimbledon High School) provided us with a nice explanation:

"Diagram 2 (D.2) shows the shape reflected across its horizontal base.

Diagram 3 (D.3) is Diagram 2 rotated 90° clockwise. You can see that the angle of the tip is 30°+30°, adding up to 60°. If a line is drawn between the bottom two vertices as shown in blue, that line should be 12cm because the other two sides are 12cm and the angle between them is 60°(making the triangle an equilateral triangle). The blue line drawn should form an isosceles triangle with the two 5cm lines. In an isosceles triangle, the length of the two identical sides must add up to greater than the length of the base. In this particular isosceles triangle, the identical sides add up to 10cm which is shorter than the base of 12cm. Thus, the triangle in the problem is impossible to construct!"

Giancarlo used trigonometry to solve the problem. By extending the original triangle to make a right-angled triangle, he found that the length of the new, shorter side was 12sin30 cm, which is 6cm, and so there is no solution for the same reason as above.