Complete the hexagonal and star shaped Rangolies [see Fig. (i) and (ii)] by filling them with as many equilateral triangles of side $1$ cm as you can. Count the number of triangles in each case. Which has more triangles?

Question

Complete the hexagonal and star shaped Rangolies [see Fig. (i) and (ii)] by filling them with as many equilateral triangles of side

1

cm as you can. Count the number of triangles in each case. Which has more triangles?

Filo · Accepted Answer

(i) From the figure, we can say that the rangoli is in the shape of a regular hexagon.

Hence,

6

equilateral triangles each of side

5

cm, can be drawn in it.

A (Δ PQR) = \frac{3}{4} (s i d e)^{2} = \frac{3}{4} \times 5^{2}

A (Δ PQR) = \frac{25 3}{4} c m^{2}

∴ A (R an g o l i) = 6 \times A (Δ PQR) = \frac{150 3}{4} c m^{2}

Area of equilateral triangle of side

1 c m = \frac{3}{4} (1)^{2} = \frac{3}{4} c m^{2}

\mbox {No. of equilateral triangles in rangoli} = \dfrac {A(Rangoli)}{A( eq. \Delta of 1cm)} = \dfrac {\frac {150\sqrt 3}{4}}{\frac {\sqrt 3}{4}} = 150

There can be

150

equilateral triangles each of side

1

cm in the hexagonal rangoli.

(ii) From the figure, we can say that the rangoli is in the shape of a star.

Hence,

12

equilateral triangles each of side

5

cm, can be drawn in it.

∴ A (R an g o l i) = 12 \times \frac{3}{4} (5)^{2} = 753 c m^{2}

Area of equilateral triangle of side

1 c m = \frac{3}{4} (1)^{2} = \frac{3}{4} c m^{2}

\mbox {No. of equilateral triangles in rangoli} = \dfrac {A(Rangoli)}{A( eq. \Delta of 1cm)} = \dfrac { 75\sqrt 3}{\frac {\sqrt 3}{4}} = 300

There can be

300

equilateral triangles each of side

1

cm in the hexagonal rangoli.

Hence, star shaped rangoli has more equilateral triangles in it.

Question Text	Complete the hexagonal and star shaped Rangolies [see Fig. (i) and (ii)] by filling them with as many equilateral triangles of side $1$ cm as you can. Count the number of triangles in each case. Which has more triangles?
Answer Type	Text solution:1
Upvotes	150

Complete the hexagonal and star shaped Rangolies [see Fig. (i) and (ii)] by filling them with as many equilateral triangles of side $1$ cm as you can. Count the number of triangles in each case. Which has more triangles?

Text solutionVerified

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