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Question:

Complete the hexagonal and star shaped Rangolies by filling them with as many equilateral triangles of side 1cm as you can. Count the number of triangles in each case. Which has more triangles?

Solution:

(i)From the figure, we can say that the rangoli is in the shape of a regular hexagon.Let the area of hexagon be P = 3√3/2(side)² = 3√3/2 × 5² P = 75√3/2 cm² ∴A(Rangoli) = 75√3/2 cm² Let area of equilateral triangle of side 1cm be A′ A′ = √3/4(1)² = √3/4 cm² Let no. of equilateral triangles in rangoli be n = A(Rangoli)/A(eq.Δof1cm) = (75√3/2)/(√3/4) = 150 There can be 150 equilateral triangles each of side 1cm in the hexagonal rangoli (ii)From the figure, we can say that the rangoli is in the shape of a star.Hence, the figure consists of 12 equilateral triangles each of side 5cm. ∴A(Rangoli) = 12 × √3/4(5)² = 75√3 cm² Let area of equilateral triangle of side 1cm be A′ A′ = √3/4(1)² = √3/4 cm² No. of equilateral triangles in rangoli = A(Rangoli)/A(eq.Δof1cm) = 75√3/(√3/4) = 300 There can be 300 equilateral triangles each of side 1cm in the hexagonal rangoli.Hence, star shaped rangoli has more equilateral triangles in it.