case study-2 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 given 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 only
(b)congruent only
(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 6 units,
(a) 2 sq units
(b) 6 ×root3 sq units
(c) 47×root3 sq units
(d) 9×root3 sq units
v) find the area of hexagon, if each edge is 6 units.
(a) 12 sq units maths basic
(b) 54×root3 sq units
(c) 36×root3 sq units
(d) 24×root3 sq units
Answers
Answer:
Step-by-step explanation:
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.
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.)
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.
iv) find the area of equilateral triangle if each edge is 6 units,
(a) 2 sq units
(b) 6 ×root3 sq units
(c) 47×root3 sq units
(d) 9×root3 sq units
= area = √3 a² /4
= √3× 6² /4
=√3 × 36/4
=√3 × 9
= 9√3 sq.unit
v) find the area of hexagon, if each edge is 6 units.
(a) 12 sq units maths basic
(b) 54×root3 sq units
(c) 36×root3 sq units
(d) 24×root3 sq units
= Area of one equilateral triangle is 9√3.
= In hexagon, it has 6 equilateral triangle, so the area of hexagon will be
= 6× 9√3
= 54√3 sq unit
Question 1) :- find the number of equilateral triangles in the given figure.
(a) 6 (b) 4 (c) 3 (d) 8
Solution :-
given that, a regular hexagon is divided into equaliteral ∆'s.
so,
- Total number of equilateral triangles = 6 (a) (Ans.)
Question 2) :- 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
Solution :-
when the areas of two equilateral triangles are equal,
- They have equal sides .
so,
- Both ∆'s are Congruent only (b) (Ans.)
Question 3) :- How many triangles are similar in the given figure ?
(a) 6 (b) 4 (c) 3 (d) 8
Answer :-
As we can see that, all the sides of the hexagon is equal so 6 equilateral are formed .
therefore, 6 (a) ∆'s are similar in the given figure.
Question 4) :- find the area of equilateral triangle if each edge is 6 units ?
(a) 2 sq units
(b) 6 ×root3 sq units
(c) 47×root3 sq units
(d) 9×root3 sq units
we know that,
- Area of equilateral triangle = (√3/4) * (side)²
so,
→ Area = (√3/4) * (6)²
→ Area = (√3/4) * 36
→ Area = 9√3 sq units. (d) (Ans.)
Question 5) :- Find the area of hexagon, if each edge is 6 units.
(a) 12 sq units .
(b) 54×root3 sq units
(c) 36×root3 sq units
(d) 24×root3 sq units
Solution :-
we know that,
- Area of regular hexagon = 6 * (Area of Equaliteral ∆)
so,
→ Req. Area = 6 * (9√3)
→ Re. Area = 54√3 sq units (b) (Ans.)
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