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Q.

The apothem of a square having its area numerically equal to its perimeter is compared to the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:


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a

Equal to the second

b

4 3  times the second

c

2 3  times the second

d

2 3   times the second 

answer is A.

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Detailed Solution

Let’s first consider the square.
Let each side of a square be  S 1   and let the apothem of a square be  A 1  
 Area of a square = S 1 2  
Perimeter of square =4 S 1  
According to the question, the numerical value of the area of the square is equal to its perimeter.
Therefore,
S 1 2 =4 S 1  .
Subtracting 4 S 1   from S 1 2  , we get
S 1 2 −4 S 1 =0  
On further simplification, we get
S 1 ( S 1 −4)=0 S 1 =4 &  S 1 ≠0  
Thus, the side of a square is 4 units.
We know the relation between the apothem of a square and its sides.
2 A 1 = S 1  
Putting the value of S 1  , we get
2 A 1 =4 A 1 =2  
We got the value of a square.
Now, we will find the value of the apothem of a given equilateral triangle.
Let each side of an equilateral triangle be  S 2  , and let the apothem of the equilateral triangle be  A 2  
Area of the equilateral triangle = 3 4 S 2 2   
Perimeter of equilateral triangle =3 S 2  
According to the question, the numerical value of the area of an equilateral triangle is equal to its perimeter.
Therefore,
3 4 S 2 2 =3 S 2  
Subtracting 3 S 2   from 3 4 S 2  , we get
3 4 S 2 2 −3 S 2 2 =0 S 2 ( 3 4 S 2 −3)=0 S 2 =4 3 &  S 2 ≠0  
Thus, the side of the equilateral triangle is 4 3  units.
We know the relation between the apothem of an equilateral triangle and its sides.
2 3 A 2 = S 2  
Putting the value of S 2  , we get
2 3 A 2 =4 3 A 2 =2  
We got the value of the apothem of the equilateral triangle.
The apothem of square is equal to the apothem of the equilateral triangle.
A 1 = A 2  
 
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