Three sides of a triangle are 18 cm, 20 cm and34 cm Find the area of a rectanglearea that can be inscribed in this triangle on| Side 34 cm as a base.The 3 sides of a triangle are 18 cm, 20 cm and 34 cm.| The maximum size imposed within that triangleExplain the area of which the base is on the 34 cm arm -
Answers
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