Find the area of a rhombus whose perimeter is 200 m and one of the diagonals is 80 m.

Let each of the equal sides of the rhombus be a cm. Then,
Perimeter = a + a + a + a = 4a m According to the question,

                4a = 200

rightwards double arrow space space space space space space space space space space space space space space space straight a equals 200 over 4 equals 50 space straight m


Let each of the equal sides of the rhombus be a cm. Then,Perimeter =

straight d subscript 1 equals 80 space straight m
straight a squared space equals space open parentheses straight d subscript 1 over 2 close parentheses squared plus open parentheses straight d subscript 2 over 2 close parentheses squared
rightwards double arrow space space left parenthesis 50 right parenthesis squared equals left parenthesis 40 right parenthesis squared plus open parentheses straight d subscript 2 over 2 close parentheses squared
rightwards double arrow space space open parentheses straight d subscript 2 over 2 close parentheses squared equals left parenthesis 30 right parenthesis squared
rightwards double arrow space space straight d subscript 2 over straight d equals 30
rightwards double arrow space space straight d subscript 2 equals 60 space straight m

∴ Area of the rhombus

equals 1 half straight d subscript 1 straight d subscript 2 equals 1 half cross times 80 cross times 60
equals space 2400 space straight m squared

646 Views

he perimeter of a rhombus is 146 cm. One of its diagonals is 55 cm. Find the length of the other diagonal and area of the rhombus.

Length of a side of the rhombus

equals 146 over 4 cm equals 36.5 space space cm.



Length of a side of the rhombusFor ΔABCa = 36.5 cm b = 55 cm c = 36.

For ΔABC
a = 36.5 cm b = 55 cm c = 36.5 cm

therefore space space space space space space space 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 36.5 plus 55 plus 36.5 over denominator 2 end fraction equals 128 over 2
space space space space space space space space space space space space space space equals 64 space space cm

Area of the ΔABC

equals square root of straight s left parenthesis straight s minus straight a right parenthesis asterisk times straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root
equals space square root of 64 left parenthesis 64 minus 36.5 right parenthesis left parenthesis 64 minus 55 right parenthesis left parenthesis 64 minus 36.5 right parenthesis end root
equals square root of 64 left parenthesis 27.5 right parenthesis left parenthesis 9 right parenthesis left parenthesis 27.5 right parenthesis end root
equals space 8 cross times 27.5 cross times 3
equals space 660 space cm squared

∴ Area of the rhombus ABCD
= 2 Area of the ΔABC = 2 x 660 = 1320 cm2


rightwards double arrow space space space space space 1 half straight d subscript 1 straight d subscript 2 equals 1320
rightwards double arrow space space space space 1 half left parenthesis 55 right parenthesis straight d subscript 2 equals 1320
rightwards double arrow space space space space space space straight d subscript 2 equals fraction numerator 1320 cross times 2 over denominator 55 end fraction
rightwards double arrow space space space space space space straight d subscript 2 equals 48 space cm

∴ Length of the other diagonal is 48 cm. 

998 Views

The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm and the diagonal AC measures 42 cm. Find the area of the parallelogram.  

For ΔABC,   a = 34 cm b = 42 cm c = 20 cm


For ΔABC,   a = 34 cm b = 42 cm c = 20 cm∴ Area of ΔABC
∴ Are

therefore space space space straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction space equals fraction numerator 34 plus 42 plus 20 over denominator 2 end fraction equals 48 space cm

∴ Area of ΔABC

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 space square root of 48 left parenthesis 48 minus 34 right parenthesis left parenthesis 48 minus 42 right parenthesis left parenthesis 48 minus 20 right parenthesis end root
equals space square root of 48 left parenthesis 14 right parenthesis left parenthesis 6 right parenthesis left parenthesis 28 right parenthesis end root
equals space 336 space cm squared

∴ Area of parallelogram ABCD
= 2 Area of triangle ABC
= 2 x 336 cm2 = 672 cm2



1252 Views

Advertisement

A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 15 cm, 14 cm and 13 cm and the parallelogram stands on the base 15 cm, find the height of the parallelogram.


For triangle
a = 15 cm b = 14 cm c = 13 cm

therefore space space space space space space space space 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 15 plus 14 plus 13 over denominator 2 end fraction equals 21 space cm
therefore space space space Area 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 space space space space space space space space equals space square root of 21 left parenthesis 21 minus 15 right parenthesis left parenthesis 21 minus 14 right parenthesis left parenthesis 21 minus 13 end root
space space space space space space space space space space space space space equals square root of 21 left parenthesis 6 right parenthesis left parenthesis 7 right parenthesis left parenthesis 8 right parenthesis end root
space space space space space space space space space space space space space equals space 84 space cm squared

Let the height of the parallelogram be h cm. Then, area of the parallelogram
= Base x Height = 15 x h = 15h cm2 According to the question,
Area of the parallelogram

                   = Area of the triangle

rightwards double arrow space space space space 15 straight h space equals space 84
rightwards double arrow space space space space space space straight h space equals 84 over 15 equals 5.6 space cm

Hence, the height of the parallelogram is 5.6 cm.





1107 Views

Advertisement
Find the area of a quadrilateral ABCD whose sides in metres are 9, 40, 28 and 15 respectively and the angle between first two sides is a right angle. 

For ΔABC Area of right triangle ABC


equals 1 half cross times Base cross times Height
equals 1 half cross times 9 cross times 40 equals 180 space straight m squared


For ΔABC Area of right triangle ABCFor ΔACDa = 28 m b = 41 m c = 15

For ΔACD
a = 28 m b = 41 m c = 15 m

therefore space space space straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction equals fraction numerator 28 plus 41 plus 15 over denominator 2 end fraction equals 84 over 2 equals 42 space straight m
therefore space space Area space of space increment ACD 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
space space space space space space space space space space space space space equals square root of 42 left parenthesis 42 minus 28 right parenthesis left parenthesis 42 minus 41 right parenthesis left parenthesis 42 minus 15 right parenthesis end root
space space space space space space space space space space space space equals square root of 42 left parenthesis 14 right parenthesis left parenthesis 1 right parenthesis left parenthesis 27 right parenthesis end root
space space space space space space space space space space space space equals space 14 cross times 3 cross times 3 equals 126 space straight m squared

Area of the quadrilateral ABCD
= Area of ΔABC + Area of ΔACD
= 180 m+ 126 m2 = 306 m2.

2649 Views

Advertisement