CASE STUDY
A beehive is an enclosed cell structure in which some honeybee species of the
subgenus Apis live and raise their young. Each cell is the form of hexagonal
shape. In a regular hexagon, there are six edges of equal lengths. Taking O as
centre, join all the vertices with the centre.

Similarity of Triangles
Two triangles are said to be similar if
- Corresponding angles are equal and
- Corresponding sides are proportional.
(i) Find the number of equilateral triangles in the corresponding figure.
(a) 6 (b) 4
(c) 3 (d) 8

(ii) If the areas of two equilateral triangles are equal, then these are always
(a) Similar
(b)Congruent
(c) Both similar and congruent
(d)None of these

(iii) How many triangles are similar in the given figure
(a) 6 (b) 4
(c) 3 (d) 8

(iv) Find the area of equilateral triangle if each edge is 4 units.
(a) 3 sq units (b) 6√3 sq units
(c ) 4√3 sq units (d) 5√3 sq units

(v) Find the area of hexagon, if each edge is 4 units.
(a) 18 sq units (b) 30√3 sq units
(c) 36√3 sq units (d) 24√3 sq units

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1) 

i) find the number of equilateral triangles in the given figure.
(a) 6 (b) 4 (c) 3 (d) 8
= 6 because all the sides of the hexagon is equal so that they are equilateral.

2) ii) if the areas of two equilateral triangles are equal, then these are always
(a) similar only
(b)congruent only
(c) both similar and congruent
(d)none of these
= Congruent only ( they are equal sides with equal area.)

3) 

iii) how many triangles are similar in the given figure
(a) 6
(b) 4
(c) 3
(d) 8
= 6 because all the sides of the hexagon is equal so that they are equilateral.

4) area = √3 a² /4
= √3×  4² /4
=√3 × 16/4
=√3 × 4
= 4√3 sq.unit

5) Area of one equilateral triangle is 4√3
in hexagon, it has 6 equilateral triangle, so the area of hexagon will be
= 6× 4√3
= 24√3 sq unit

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