If we complete the hexagonal and star-shaped rangoli [see figures] by filling them with as many equilateral triangles of side 1 cm as you can. Then which shape has more triangles?

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Hexagonal shaped rangoli

Star shaped rangoli

Both have equal numbers of triangles

None of these

answer is B.

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Detailed Solution

Given that the length of each side of the hexagon is 5 cm.
It can be observed that the hexagonal-shaped rangoli has 6 equilateral triangles of each side 5 cm in it. Complete the hexagonal and star shaped Rangolies [see Fig. 7.53 (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?

Complete the hexagonal and star shaped Rangolies [see Fig. 7.53 (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?

Let's find the area of

Δ O A B .

We know that area of an equilateral triangle of side 'a' is given by

34 × a 2 .

Therefore, area of ΔOAB is,

⇒ 34 × 52 ⇒ 34 × 25

Therefore, area of the hexagonal-shaped rangoli is,

⇒ 6 × 2534 cm 2

⇒ 3 × 2532 cm 2

⇒ 7532 cm 2

Now, area of an equilateral triangle of side 1 cm is,

⇒ 34 × 12

⇒ 34 cm 2

Therefore, number of equilateral triangles of 1 cm side that can be filled in this hexagonal shaped Rangoli =

Area of hexagonal-shaped rangoli Area of an equilateral triangle of side 1 cm

⇒ 753234

⇒ 753 × 4 2 × 3 ⇒ 150

Now, the star-shaped rangoli has 12 equilateral triangles of side 5 cm in it.
Complete the hexagonal and star shaped Rangolies [see Fig. 7.53 (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?

Area of star-shaped rangoli

= 12 × 2534 cm 2 .

⇒ 753 cm 2

Therefore, number of equilateral triangles of 1 cm side that can be filled in this star-shaped rangoli =

Area of star-shaped rangoli Area of an equilateral triangle of side 1 cm

⇒ 75334

⇒ 753 × 4 3 ⇒ 300

Here, the number of equilateral triangles of 1 cm side that can be filled in the star-shaped Rangoli is 300 and the the number of equilateral triangles of 1 cm side that can be filled in hexagonal-shaped Rangoli is 150.
Therefore, star-shaped rangoli has more equilateral triangles in it.
Hence, option (2) is correct.

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