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Sides of a triangle are in the ratio of 12 :17 : 25 and its perimeter is 540 cm. Find its area.

Let the sides of the triangle be 12k, 17k and 25 k cm.
Then,
Perimeter = 12k+ 17k + 25k = 54k
According to the question,
54k = 540

rightwards double arrow space space space space space straight k equals 540 over 54

rightwards double arrow   k = 10

therefore   a = 12k = 12 x 10 = 120 cm
       b = 17k = 17 x 10 = 170 cm
       c = 25k = 25 x 10 = 250 cm

therefore space space space space straight s space equals space fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction equals fraction numerator 120 plus 170 plus 250 over denominator 2 end fraction
space space space space space space space space space equals space 540 over 2 equals 270 space cm
therefore space space space Area space space equals space square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root
space space space space space equals space square root of 270 left parenthesis 270 minus 120 right parenthesis left parenthesis 270 minus 120 right parenthesis left parenthesis straight v minus 250 right parenthesis end root
space space space space space equals square root of 270 left parenthesis 150 right parenthesis left parenthesis 100 right parenthesis left parenthesis 20 right parenthesis end root
space space space space space equals square root of left parenthesis 9 cross times 30 right parenthesis left parenthesis 5 cross times 30 right parenthesis left parenthesis 5 cross times 20 right parenthesis left parenthesis 20 right parenthesis end root
space space space space space equals space left parenthesis 3 right parenthesis left parenthesis 30 right parenthesis left parenthesis 5 right parenthesis left parenthesis 20 right parenthesis space cm squared space equals space 9000 space cm squared.

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The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see figure). The advertisements yield an earning of र5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?


a =122 m
b = 22 m
c = 120 m

therefore space space space space straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction equals fraction numerator 122 plus 22 plus 120 over denominator 2 end fraction straight m
space space space space space space space space equals space 264 over 2 space space straight m space equals space 132 space straight m

 
a =122 mb = 22 mc = 120 m 1 year =12 monthsV Rent for 12 months per

1 year =12 months
V Rent for 12 months per m2= र 5000

therefore space   Rent for 1 month per m2 र5000 over 2

therefore   Rent for 3 months of 1320 m2

                equals space र 5000 over 12 cross times 3 equals space र space 1250

therefore space space  Rent for 3 months of 1320 m2

            = र (1250 x 1320)
            = र 1650000.

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There is a slide in a park. One of its side walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN”. If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.


a = 15 m 
b = 11 m 
c = 6 m

therefore space space space space space straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction
space space space space space space space space space equals space fraction numerator 15 plus 11 plus 6 over denominator 2 end fraction straight m space equals space 16 space straight m

therefore    Area painted in colour

    equals square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b left parenthesis straight s minus straight c right parenthesis end root
equals square root of 16 left parenthesis 16 minus 15 right parenthesis left parenthesis 16 minus 11 right parenthesis left parenthesis 16 minus 6 right parenthesis end root space straight m squared
equals space square root of 16 left parenthesis 1 right parenthesis left parenthesis 5 right parenthesis left parenthesis 10 right parenthesis end root space straight m squared space equals space 20 square root of 2 space straight m squared.

3494 Views

Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm. 

 a = 18 cm
b = 10 cm Perimeter - 42 cm
⇒ a + b + c = 42
⇒ 18 + 10 + c = 42
⇒ 28 + c = 42
⇒    c = 42 – 28
⇒    c = 14 cm

 space space space space space space space space space space space space space straight s equals 42 over 2 equals 21 space cm

therefore space space space space space Area of the triangle 

       equals square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root
equals square root of 21 left parenthesis 21 minus 18 right parenthesis left parenthesis 21 minus 10 right parenthesis left parenthesis 21 minus 14 right parenthesis end root
equals square root of 21 left parenthesis 3 right parenthesis left parenthesis 11 right parenthesis left parenthesis 7 right parenthesis end root
equals space square root of left parenthesis 7 right parenthesis left parenthesis 3 right parenthesis left parenthesis 3 right parenthesis left parenthesis 11 right parenthesis left parenthesis 7 right parenthesis end root
equals space left parenthesis 7 right parenthesis left parenthesis 3 right parenthesis square root of 11 equals 21 square root of 11 space cm squared

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A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?


'a'   =  a
'b'   =  a
'c'   =  a

therefore space space space space space space space space straight s space equals space fraction numerator apostrophe straight a apostrophe plus apostrophe straight b apostrophe plus apostrophe straight c apostrophe over denominator 2 end fraction equals fraction numerator straight a plus straight a plus straight a plus over denominator 2 end fraction equals fraction numerator 3 straight a over denominator 2 end fraction

Area of the signal board


equals square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root
equals square root of fraction numerator 3 straight a over denominator 2 end fraction open parentheses fraction numerator 3 straight a over denominator 2 end fraction minus straight a close parentheses open parentheses fraction numerator 3 straight a over denominator 2 end fraction minus straight a close parentheses open parentheses fraction numerator 3 straight a over denominator 2 end fraction minus straight a close parentheses end root
square root of fraction numerator 3 straight a over denominator 2 end fraction open parentheses straight a over 2 close parentheses open parentheses straight a over 2 close parentheses open parentheses straight a over 2 close parentheses end root equals square root of fraction numerator 3 straight a to the power of 4 over denominator 16 end fraction end root equals fraction numerator square root of 3 over denominator 4 end fraction straight a squared

Perimeter = 180 cm

rightwards double arrow   'a' + 'b' + 'c' = 180
rightwards double arrow   a + a + a = 180
rightwards double arrow              3a = 180

rightwards double arrow space space space space space space space straight a space equals space 180 over 3
rightwards double arrow space space space    a = 60  cm

therefore  Area of the signal board equals fraction numerator square root of 3 over denominator 4 end fraction straight a squared

           equals fraction numerator square root of 3 over denominator 4 end fraction left parenthesis 60 right parenthesis squared equals 900 square root of 3 space cm squared
Alternatively,

     space space space space space straight s space equals fraction numerator 3 straight a over denominator 2 end fraction equals 3 over 2 left parenthesis 60 right parenthesis equals 90 space cm

Area of the signal board

    equals square root of straight s left parenthesis straight s minus straight a left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root
equals square root of 90 left parenthesis 90 minus 60 right parenthesis left parenthesis 90 minus 60 right parenthesis left parenthesis 90 minus 60 right parenthesis end root
equals space square root of 90 left parenthesis 30 right parenthesis left parenthesis 30 right parenthesis left parenthesis 30 right parenthesis end root equals 900 square root of 3 space cm squared


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