Triangle ABC has sides Z, √Z and Z^2, where Z is an integer. What is t

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Competition Mode Question

Triangle ABC has sides  Z ,  √Z  and  Z^2 , where  Z  is an integer. What is the area of ABC?

A.  √2

B.  2

C.  2√2

D.  \frac{√3}{4}

E.  \frac{√15}{2}

ArtVandaley · Accepted Answer

Ah an Excellent question.

The catch here is that, for no values of Z except 1,a triangle is possible.

Lets take the case of Z=2, the sides will be 2 , root2 and 4. Now third side is longer than the sum of other two sides, which is not possible.

Hence Z=1. Triangle is equilateral. And area is root3/4.

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EncounterGMAT · Answer

Let Z=1
If a side Z=1, then √Z=1 and Z^2=1....which is nothing but equilateral triangle.

Area of equilateral triangle ABC= √3/4 * side^2 = √3/4 * 1^2 = √3/4

Imo answer is option D

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Mohammadmo · Answer

Let Z,1
p=1+1+1=3
According to Heron's formula, we have 
P=√p/2*(p/2-1)*(p/2-1)*(p/2-1)=
√3/2*1/2*1/2*1/2=√3/16= √3 /4
Option D

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reach2sangram · Answer

Z=1 
So triangle will be equilateral
Answer is √3/4 D

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unraveled · Answer

Since Z is an integer only one positive value(length can’t be negative) is possible. Thus,  Z = 1  and  √Z = 1  and  Z^2 = 1  as  Z = 2  would not make a triangle because other two side would be  √Z = √2  and  Z^2 = 4 . Here  Z + √Z < Z^2  where sum of two smaller sides must be greater than the largest side.

Hence the triangle has each side equal to 1. So, it’s an equilateral triangle.

Therefore Area of equilateral triangle  = \frac{√3}{4} * side^2 .
  = \frac{√3}{4}

Also, using Hero’s formula 
 S = \frac{(a + b + c)}{2}  where a, b and c are three sides of triangle.
 S = \frac{3}{2}

Area of triangle  = √(s(s-a)(s-b)(s-c)) 
  = √(\frac{3}{2} * (\frac{3}{2}-1) * (\frac{3}{2}-1) * (\frac{3}{2}-1)) 
  = √(\frac{3}{2} * (\frac{1}{2})^3) 
  = \frac{√3}{4}

Answer (D).

madgmat2019 · Answer

if we check the sum of sides property

z+z^2 > sqrt(z) for many values
also sqrt(z)+z^2 >z for many values
 but z+sqrt(z)>z^2 only if z=1

so area is (sqrt(3)*z)/4 ......  OA: D

Peddi · Answer

According to given sides triangle ABC is possible only when all sides are equal I.e when z=√z=z^2, it happens when z=1 , for any other integer third side rule does not satisfy, so since all sides are equal area of equilateral triangle is√3/4 a^2 = √3/4

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Triangle ABC has sides Z, √Z and Z^2, where Z is an integer. What is t

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Triangle ABC has sides Z, √Z and Z^2, where Z is an integer. What is t

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