Math, asked by aryansahani1920, 1 year ago

three sides of a triangle are 18 cm 20 cm and 34 cm find the area of a rectangle of maximum area that can be inscribed in this triangle on side 34 CM as a base​

Answers

Answered by amitnrw
0

Answer:

72  cm²

Step-by-step explanation:

Triangle has three sides

18 cm, 20 cm  & 34 cm

s =  ( 18 + 20 + 34)/2 = 36

Area using Hero formula

Area = √36(36-18)(36-20)(36-34) = 144 cm²

Area = (1/2)Base * Altitude =  (1/2) * 34 * altitude = 144

=> Altitude = 144/17

Altitude splits 34 into √18² - (144/17)²   & √20² - (144/17)²

=  270/17   &  308/17 cm

Let say Height of rectangle = y and other side= x ( x = x1 + x2)

x1 is toward  270/17 cm  side  & x2 is toward 308/17 cm

Similar triangle are formed by height of rectangle

=> Altitude / y   =   (270/17)/ ((270/17) - x1)

=> (144/17) / y =  (270/17)/ ((270/17) - x1)

=>  144 ((270/17) - x1)  = 270y

=> 144*270  - 144*17x1 = 270*17 y

=> x1  = 270( 144 - 17y) / (144 * 17)

=> Altitude / y   =   (308/17)/ ((308/17) - x2)

=> (144/17) / y =  (308/17)/ ((308/17) - x2)

=>  144 ((308/17) - x2)  =308y

=> x2 = 308(144 - 17y)/ (144 * 17)

x1/x2 =270/308 = 135/154

x1 + x2 = 578 * (144 - 17y)/ (144 * 17)  =  (17/72) (144 - 17y)

Area of rectangle = (x1 + x2)y

=  (17/72) (144 - 17y)y

= 34y  - 289y²/72

Taking differentiation

dA/dy =  34 - 289y/36

Equating with 0

34 - 289y/36 = 0

=> y = 34 * 36/289

=> y = 72/17

Putting this value of y

Area =  (17/72) ( 144  - 72) (72/17)

= 72 cm²

Max Area of rectangle = 72  cm²

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