A traffic signal board is equilateral in shape, with words ‘SCHOOL AHEAD ‘with on it. The perimeter of the board is 180 cm and then find the area of the signal board .
Answers
The perimeter of a triangle is equal to the sum
of its three sides it is denoted by 2S.
2s=(a+b+c)
s=(a+b+c)/2
Here ,s is called semi perimeter of a triangle.
The formula given by Heron about the area of a
triangle is known as Heron's formula. According to this formula area of a
triangle= √s (s-a) (s-b) (s-c)
Where a, b and c are three sides of a triangle
and s is a semi perimeter.
This formula can be used for any triangle to
calculate its area and it is very useful when it is not possible to find the height
of the triangle easily . Heron's formula is
generally used for calculating area of scalene triangle.
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Solution:
Given, side of a
signal whose shape is an equilateral triangle= a
Semi perimeter, s=a+a+a/2=
3a/2
Using heron’s
formula,
Area of the signal board = √s (s-a) (s-b) (s-c)
= √(3a/2) (3a/2 – a) (3a/2 – a) (3a/2 –
a)
= √3a/2 × a/2 × a/2 × a/2
= √3a⁴/16
= √3a²/4
Hence, area of signal board with
side a by using Herons formula is= √3a²/4
Now,
Perimeter of an equilateral
triangle= 3a
Perimeter of the traffic signal board = 180 cm (given)
3a = 180 cm
a = 180/3= 60 cm
Now , area of
signal board=√3a²/4
= √3/4 × 60 × 60
= 900√3 cm²
Hence , the area
of the signal board when perimeter is 180 cm is 900√3 cm².
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Hope this will help you....
If the board is a square:
perimeter = 180cm
1 side = 180 ÷ 4
= 45cm
area = 45 × 45
= 2025cm²