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Two points are drawn on each side of a square with an area of 81 square units dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your answer as a decimal to the nearest tenth.

tertre  Jul 23, 2017

#1
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The area of the square is 81 square units.

81 un2  =  (side length)2

9 un   =   side length

The two points divide the side length into 3 congruent parts.

So, each piece will be  9/3  =  3  units long.  Now we know...

radius of each quarter circle  =  3 un             and...

length of straight piece  =  3 un

On the shape's perimeter...there are  4  quarter circles, and there are 4  straight pieces.  So...

perimeter  =  4[ circumference of quarter circle ]  +  4[ length of straight piece ]

=  4[ (2π * radius)/4 ]  +  4[ length of straight piece ]

=  [ 2π * radius ] + 4[ length of straight piece ]

=  [ 2π * 3 ] + 4[ 3 ]

=  6π + 12

≈  30.8 un

hectictar  Jul 23, 2017
Sort:

#1
+5541
+1

The area of the square is 81 square units.

81 un2  =  (side length)2

9 un   =   side length

The two points divide the side length into 3 congruent parts.

So, each piece will be  9/3  =  3  units long.  Now we know...

radius of each quarter circle  =  3 un             and...

length of straight piece  =  3 un

On the shape's perimeter...there are  4  quarter circles, and there are 4  straight pieces.  So...

perimeter  =  4[ circumference of quarter circle ]  +  4[ length of straight piece ]

=  4[ (2π * radius)/4 ]  +  4[ length of straight piece ]

=  [ 2π * radius ] + 4[ length of straight piece ]

=  [ 2π * 3 ] + 4[ 3 ]

=  6π + 12

≈  30.8 un

hectictar  Jul 23, 2017

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