The area of a rhombus is 2016 sq cm and its side is 65 cm. The lengths of the diagonals (in cm) respectively are ??
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The area of a rhombus is 2016 sq cm and its side is 65 cm. The lengths of the diagonals (in cm) respectively are ??
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it's Isha
ask so me
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ANSWER:
126 cm and 32 cm
EXPLAINATION
Let ABCD be the given rhombus, AB, BC, CD and DA are the sides of rhombus . let the diagonals intersect each other at point O. Since the diagonal of rhombus bisect each other so, OA = OC and OB = OD.
Let the length of OC = x cm
Let the length of OB = y cm
We have given the length of side of rhombus = 65cm
Also Area of rhombus = 2016sq. cm…………(i)
Area of rhombus = 12×AC×DB
=12×(AO+OC)×(BO+OD)
=12×2x×2y=2xy
So, Area of the rhombus = 2xy ………..(ii)
Equating (i) and(ii) we get
2xy = 2016 ………..(iii)
⇒xy=1008 ……………(iv)
Now form the figure BOC is a right angled triangle at B. so, by Pythagoras theorem
BC2=OB2+OC2
BC is the side of rhombus, os BC = 65 OB = y and OC = x
∴652=y2+x2
x2+y2=4225 …………(v)
Adding equation (iii) and (v) we get
x2+y2+2xy=2016+4225
∴(x+y)2=6241 { using identity x2+y2+2xy=(x+y)2 }
x+y=6241−−−−√
x+y=79 ……………….(vi)
Now from equation (iv) we can find y from x
∴y=1008x …………….(vii)
Putting value of equation (vii) in equation (vi)
x+1008x=79
x2+1008x=79
x2+1008=79x
x2−79x+1008=0
Applying quadratic formula
x=79±(79)2−4×1×1008−−−−−−−−−−−−−−−−√2
=79±6241−4032−−−−−−−−−−√2
=79±2209−−−−√2
x=79±472
⇒x=79+472,x=79−472
x=1262,322
x=63,16
So, when x = 63, value of y = 100863=16
When x = 16, value of y =100816=63
We get OC = 63 and OD = 16
AC= OA+OC = 63+63 = 126
BD = OB+OD = 16+16 = 32
So, diagonals of the rhombus are 126 cm and 32 cm
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