Towers

562 /www/nrich/html/content/98/03/six5/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <mdo:image src="towers.gif" alt=""></mdo:image> We build an imaginary tower of squares inside a right angled isosceles triangle. The largest square stands on the hypotenuse of the right angled triangle. Each square has two vertices touching the other sides of the triangle. Only three squares are drawn in the diagram but imagine that there are infinitely many getting smaller and smaller and smaller... What fraction of the area of the triangle is covered by the squares? You can do this without a lot of calculation and without any advanced mathematics. If you wish to extend this project you can ask: What if the triangle was equilateral? Or what if the tower was made up of rectangles? Or... </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The first part of this problem was solved by Polly, West Flegg Middle School in Great Yarmouth; Joel, ACS in Singapore; and Luke, Dan, Nicholas and Luke, all from Clevedon Community School. They all relized that the area of the triangles on either side of the squares are half that of the square (the base angle of triangle is 45°). Therefore, each square is half of the trapezium it is contained within, and the sum of all the squares will be half of the sum of all the trapezia that make up the triangle. The area covered by the squares is half the area covered by the triangle for the right-angled isosceles triangle. But, what happens when the triangle is equilateral or just isosceles? You will need to do some calculating to work out what happens then. </mdoxml> 2 4 0 0 1 0 0 Towers A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares? Sequences, Functions and Graphs Sequences Sequences, Functions and Graphs Series Measures and Mensuration Area Numbers and the Number System Infinity