Question

length and breadth of a rectangle are 8cm and 6 cm respectively the rectangle is cut on its four vertics such that the resulting figure is a regular octagon what is the side of the octagon?? solve this question jisko aata ho wo hi answer kare faltu ke answers mt karna​

Answer 1

Answer:

perimeter of the rectangle=2(l+b)

2×(6+8)=82

sides of octagon=82/8=

=10.25

Answer 2

Let ABCD be the rectangle such that AB = CD = 8 cm and BC = DA = 6 cm.

Let the side of the regular octagon = s.

Let the 4 vertices be trimmed such x is on sides BC and DA and y is on sides AB and CD. So at the four vertices we are cutting off right angled triangles of sides x and y while their hypotenuse is s.

2y+s = 8 …(1)

2x+s = 6 …(2)

x*2+y^2 = s^2 …(3)

From (1), y = (8-s)/2 and from (2), x = (6-s)/2. Now Eq. (3) becomes

[(6-s)/2]^2 + [(8-s)/2]^2 = s^2 or

[(6-s)]^2 + [(8-s)]^2 = 4s^2 or

36–12s+s^2+64–16s+s^2 = 4s^2 or

100–28x-2s^2 = 0 or

s^2+14s-50 = 0

s = [-14 +(14^2+200)^0.5]/2

= [-14 + (196+200)^0.5′

Let ABCD is the rectangle whose AB =8 cm, BC=6cm,CD=8cm, and DA=6cm. E,F,G,H are the middle points of AB, BC, CD, and DA respectively. M,N,O,P,Q,R,S,T are the vertices of the regular octagon that formed ,and point M is on the line AE , other points N,O,P etc are on EB, BF, FC so on. Now MN=NO=OP=PQ=QR=RS=ST=TM= 2x .Then ME=x,

As AE=8/2= 4cm , AM=4-x

Similarly AT=3-x ,and TM=2x

As triangle ATM is a right triangle TM^2=AM^2+AT^2,

Or, (2x)^2=(4-x)^2+(3-x)^2

Or, 4x^2=16–8x+x^2+9–6x+x^2

Or, 2x^2+14x-25=0

Or,x=[-14±√{14^2–4×2×(-25)}]/(2×2)

Or, x={-14±√(196+200)}/4

Or, x=(-14±√396)/4

Or x=(-14±19.9)/4

Or x=5.9/4 …..Negative value discarded .

Or, x= 1.47 cm

So, ME =x=1.47 cm

MN=NO=…= 2x=2×1.47 = 2.94cm

Sides of the regular octagon is 2.94 cm

The way to solve this problem is to consider size of Octagon as x and thus (6-x)/2 would be the cut on either side of x, on the breadth of the rectangle having side 6. Similarly the cut on either side of x, on the length having side 8 would be (8-x)/2.

Now we have to find x such that ((6-x)/8)^2 + ((8-x)/2)^2 = x^2.

If we solve for x, we get the value of x to be 2.9498743710662.

Thus the required answer is 2.9498743710662.

Answer is 2.94cm