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Geometry Difficulty 2.4 Junior Find the answer

A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. How many tiles are used to form the figure in the sequence with perimeter 91 cm?

A number or a short expression. Spacing and $ signs are ignored.

Solution

The first figure consists of one tile with perimeter 3×7 cm=21 cm3 \times 7 \mathrm{~cm} = 21 \mathrm{~cm}. Each time an additional tile is added, the perimeter of the figure increases by 7 cm (one side length of a tile), because one side length of the previous figure is 'covered up' and two new side lengths of a tile are added to the perimeter for a net increase of one side length (or 7 cm). Since the first figure has perimeter 21 cm and we are looking for the figure with perimeter 91 cm, then the perimeter must increase by 91 cm21 cm=70 cm91 \mathrm{~cm} - 21 \mathrm{~cm} = 70 \mathrm{~cm}. Since the perimeter increases by 7 cm when each tile is added, then 70 cm7 cm/tile=10\frac{70 \mathrm{~cm}}{7 \mathrm{~cm} / \mathrm{tile}} = 10 tiles need to be added to reach a perimeter of 91 cm. In total, this figure will have 1+10=111 + 10 = 11 tiles.

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